For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erdős–Rényi random graph $G_{n,p}$, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window $p=1/n+\Theta (n^{-4/3})$, and (ii) the inverse of its maximum value coincides with the $\Theta (n^{-4/3})$–width of the critical window. We also prove that the maximizer is not located at $p=1/n$ or $p=1/(n-1)$, refuting a speculation of Peres.

In this work, we consider a finite-state inhomogeneous-time Markov chain whose probabilities of transition from one state to another tend to decrease over time. This can be seen as a cooling of the dynamics of an underlying Markov chain. We are interested in the long time behavior of the empirical measure of this freezing Markov chain. Some recent papers provide almost sure convergence and convergence in distribution in the case of the freezing speed $n^{-\theta }$, with different limits depending on $\theta <1,\theta =1$ or $\theta >1$. Using stochastic approximation techniques, we generalize these results for any freezing speed, and we obtain a better characterization of the limit distribution as well as rates of convergence and functional convergence.

Let $\{B_k\}_{k=1}^\infty , \{X_k\}_{k=1}^\infty $ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty $ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text {dist}} {=}\text{Ber} (p_k)$, with $\sum _{k=1}^\infty p_k=\infty $, and assume that $\{X_k\}_{k=1}^\infty $ satisfy $X_k>0$ and $\mu _k\equiv EX_k<\infty $. Let $M_n=\sum _{k=1}^np_k\mu _k$, assume that $M_n\to \infty $ and define the normalized sum of independent random variables $W_n=\frac 1{M_n}\sum _{k=1}^nB_kX_k$. We give a general condition under which $W_n\stackrel{\text {dist}} {\to }c$, for some $c\in [0,1]$, and a general condition under which $W_n$ converges weakly to a distribution from a family of distributions that includes the generalized Dickman distributions GD$(\theta ),\theta >0$. In particular, we obtain the following result, which reveals a strange domain of attraction to generalized Dickman distributions. Assume that $\lim _{k\to \infty }\frac{X_k} {\mu _k}\stackrel{\text {dist}} {=}1$. Let $J_\mu ,J_p$ be nonnegative integers, let $c_\mu ,c_p>0$ and let

$ \mu _n\sim c_\mu n^{a_0}\prod _{j=1}^{J_\mu }(\log ^{(j)}n)^{a_j},\ p_n\sim c_p\big ({n^{b_0}\prod _{j=1}^{J_p}(\log ^{(j)}n)^{b_j}}\big )^{-1}, \ b_{J_p}\neq 0, $ where $\log ^{(j)}$ denotes the $j$th iterate of the logarithm.

If \[ i.\ J_p\le J_\mu ;\\ ii.\ b_j=1, \ 0\le j\le J_p;\\ iii.\ a_j=0, \ 0\le j\le J_p-1,\ \text{and} \ \ a_{J_p}>0, \] then $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\frac 1{\theta }\text{GD} (\theta ),\ \text{where} \ \theta =\frac{c_p} {a_{J_p}}. $

Otherwise, $\lim _{n\to \infty }W_n\stackrel{\text {dist}} {=}\delta _c$, where $c\in \{0,1\}$ depends on the above parameters.

We also give an application to the statistics of the number of inversions in certain random shuffling schemes.

We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and functionals of both continuous and discrete independent random variables. For random variables admitting a continuous density, it recovers classical distance bounds based on absolute third moments, with better and explicit constants. We also apply this method to multiple stochastic integrals that can be used to represent $U$-statistics, and include linear and quadratic functionals as particular cases.

We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices, each sequence converges almost surely to a real tree in the Gromov-Hausdorff-Prokhorov sense. The limiting real trees are constructed via line-breaking the half real-line with a Poisson process having rate $(\ell +1)t^\ell dt$, for each positive integer $\ell $, and the growth of the combinatorial trees may be viewed as an inhomogeneous generalization of Rémy’s algorithm.

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\sigma $, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $\sigma $ have infinite mean. The “polymer” – of length $\sigma _N$ – is given by another renewal $\tau $, whose law is modified by the Boltzmann weight $\exp (\beta \sum _{n=1}^N \mathbf{1} _{\{\sigma _n\in \tau \}})$. Our assumption is that $\tau $ and $\sigma $ have gap distributions with power-law-decay exponents $1+\alpha $ and $1+\tilde \alpha $ respectively, with $\alpha \geq 0,\tilde \alpha >0$. There is a localization phase transition: above a critical value $\beta _c$ the free energy is positive, meaning that $\tau $ is pinned on the quenched renewal $\sigma $. We consider the question of relevance of the disorder, that is to know when $\beta _c$ differs from its annealed counterpart $\beta _c^{\mathrm{ann} }$. We show that $\beta _c=\beta _c^{\mathrm{ann} }$ whenever $ \alpha +\tilde \alpha \geq 1$, and $\beta _c=0$ if and only if the renewal $\tau \cap \sigma $ is recurrent. On the other hand, we show $\beta _c>\beta _c^{\mathrm{ann} }$ when $ \alpha +\frac 32\, \tilde \alpha <1$. We give evidence that this should in fact be true whenever $ \alpha +\tilde \alpha <1$, providing examples for all such $ \alpha ,\tilde \alpha $ of distributions of $\tau ,\sigma $ for which $\beta _c>\beta _c^{\mathrm{ann} }$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($\sigma _N=\tau _N$), and one in which the polymer length is $\tau _N$ rather than $\sigma _N$. In both cases we show the critical point is the same as in the original model, at least when $ \alpha >0$.

Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [45] for the CUE and the sine process and the asymptotic formulae derived by Deift et al. [13]. For varying weights $e^{-N \operatorname{Tr} V (\mathrm{H} )}$, in the one-cut regime, we also provide estimates for the variance of linear statistics $\operatorname{Tr} f(\mathrm{H} )$ which are valid for a rather general function $f$. In particular, this implies that the logarithm of the absolute value of the characteristic polynomials of such Hermitian random matrices converges in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations introduced in [17]. For the GUE and Jacobi ensembles, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.

We consider right-continuous peacocks, that is, families of real probability measures $(\mu _t)_{t\in [0,1]}$ that are increasing in convex order. Given a sequence of time partitions we associate the sequence of martingales characterised by the fact that they are Markovian, constant on the partition intervals $[t_k,t_{k+1}[$, and such that the transition kernels at times $t_{k+1}$ are the curtain couplings of marginals $\mu _{t_k}$ and $\mu _{t_{k+1}}$. We study the limit curtain processes obtained when the mesh of the partition tends to zero and study existence, uniqueness and relevancy with respect to the original data. For any right-continuous peacock we show there exist sequences of partitions such that a limit process exists (for the finite-dimensional convergence).

Under certain additional regularity assumptions, we prove that there is a unique limit curtain process and that it is a Markovian martingale. We first study by elementary methods peacocks whose marginals correspond to uniform distributions in convex order. In this case, the results and techniques complete the results and techniques used in a parallel work by Henry-Labordère, Tan and Touzi [9]. We obtain the same type of results for all limit curtain processes associated to a class of analytic discrete peacocks, i.e., the measures $\mu _t$ are finitely supported and vary analytically in $t$.

Finally, we give examples of peacocks and sequences of partitions such that the limit curtain process is a non-Markovian martingale.

Let $K$ be a convex body in $\mathbb{R} ^n$ and $f : \partial K \rightarrow \mathbb{R} _+$ a continuous, strictly positive function with $\int \limits _{\partial K} f(x) \mathrm{d} \mu _{\partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $\mathbb{R} ^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schütt and Werner [36]. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Lévy process killed on the first exit from the positive half-line.

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha $-stable Lévy process from the Lipschitz cone in $\mathbb{R} ^d$. We obtain the Yaglom limit for the killed stable process in the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow us to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov’s theorem.

In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are frustrated. Examples are provided for three different choices of constraints.

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty }(\mathbb{R} ^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u} {\partial t}=-(-\Delta )^{\frac{\alpha } {2}}u+u\dot W(t,x)$, where $-(-\Delta )^{\frac{\alpha } {2}}$ with $\alpha \in (0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored both in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $\alpha $-stable process. As a byproduct, we obtain the critical values for $\theta $ and $\eta $ such that $\mathbb{E} \exp \left (\theta \left (\int _0^1 \int _0^1 |r-s|^{-\beta _0}\gamma (X_r-X_s)drds\right )^\eta \right )$ is finite, where $X$ is $d$-dimensional symmetric $\alpha $-stable process and $\gamma (x)$ is $|x|^{-\beta }$ or $\prod _{j=1}^d|x_j|^{-\beta _j}$.

Let $\mathcal{M} _n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset \mathbb{R} $ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in $\mathcal M_n([0,1])$ converges in the large $n$ limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on $\mathcal{M} _n(E)$ for $E=[a,b],\,E=\mathbb{R} _+$ and $E=\mathbb{R} $, respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for $E$ being a compact interval. Rather, there is a universal family of moment sequences of which the arcsine moment sequence is one particular member. On the other hand, on the moment spaces $\mathcal{M} _n(\mathbb{R} _+)$ and $\mathcal{M} _n(\mathbb{R} )$ the random moment sequences governed by our distributions exhibit for $n\to \infty $ a universal behaviour: The first $k$ moments of such a random vector converge almost surely to the first $k$ moments of the Marchenko-Pastur distribution (half line) and Wigner’s semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.

Benjamini, Kalai and Schramm showed that a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\{x: \langle x, a \rangle \le b\}$).

We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \{-1, 1\}^n \to \{-1, 1\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates.

Looking at random restrictions is necessary: we construct noise stable functions whose correlation with any half-space is $o(1)$. The examples further satisfy that different restrictions are correlated with different half-spaces: for any fixed half-space, the probability that a random restriction is correlated with it goes to zero.

We also provide quantitative versions of the above statements, and versions that apply for the Gaussian measure on $\mathbb{R} ^n$ instead of the discrete cube. Our work is motivated by questions in learning theory and a recent question of Khot and Moshkovitz.

We study a system of simple random walks on $\mathcal{T} _{d,n}=({\cal V}_{d,n},{\cal E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda $) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o} $. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o} $. Active particles perform independent simple random walk on the tree of length $ t \in{\mathbb N} \cup \{\infty \} $, referred to as the particles’ lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R} _t$ be the set of vertices which are visited by the process (with lifetime $t$). The susceptibility ${\mathcal S}({\mathcal T}_{d,n}):=\inf \{t:\mathcal{R} _t={\cal V}_{d,n} \} $ is the minimal lifetime required for the process to visit all sites. The cover time $\mathrm{CT} ({\mathcal T}_{d,n})$ is the first time by which every vertex was visited at least once, when we take $t=\infty $. We show that there exist absolute constants $c,C>0$ such that for all $d \ge 2$ and all $\lambda = {\lambda }_n >0$ which does not diverge nor vanish too rapidly as a function of $n$, with high probability $c \le \lambda{\mathcal S} ({\mathcal T}_{d,n}) /[n\log (n / {\lambda } )] \le C$ and $\mathrm{CT} ({\mathcal T}_{d,n})\le 3^{4\sqrt{ \log |{\cal V}_{d,n}| } }$.

We consider the two block stochastic block model on $n$ nodes with asymptotically equal cluster sizes. The connection probabilities within and between cluster are denoted by $p_n:=\frac{a_n} {n}$ and $q_n:=\frac{b_n} {n}$ respectively. Mossel et al. [27] considered the case when $a_n=a$ and $b_n=b$ are fixed. They proved the probability models of the stochastic block model and that of Erdős–Rényi graph with same average degree are mutually contiguous whenever $(a-b)^2<2(a+b)$ and are asymptotically singular whenever $(a-b)^2>2(a+b)$. Mossel et al. [27] also proved that when $(a-b)^2<2(a+b)$ no algorithm is able to find an estimate of the labeling of the nodes which is positively correlated with the true labeling. It is natural to ask what happens when $a_n$ and $b_n$ both grow to infinity. In this paper we consider the case when $a_{n} \to \infty $, $\frac{a_n} {n} \to p \in [0,1)$ and $(a_n-b_n)^2= \Theta (a_n+b_n)$. Observe that in this case $\frac{b_n} {n} \to p$ also. We show that here the models are mutually contiguous if asymptotically $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ and they are asymptotically singular if asymptotically $(a_n-b_n)^2 > 2(1-p)(a_n+b_n)$. Further we also prove it is impossible find an estimate of the labeling of the nodes which is positively correlated with the true labeling whenever $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ asymptotically. The results of this paper justify the negative part of a conjecture made in Decelle et al. (2011) [17] for dense graphs.

A point-shift $F$ maps each point of a point process $\Phi $ to some point of $\Phi $. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi $ is a partition of the support of $\Phi $ which is the discrete analogue of the stable manifold of $F$ on $\Phi $. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot $, the orbits of which are the $F$-foils of $\Phi $, and which is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.

An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty ,T)$. Given an additional $C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family of operators $L_t=\Delta _t +Z_t $ where $\Delta _t$ denotes the Laplacian with respect to the metric $g_t$. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by $L_t$, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.

We analyze the dynamics of moderate fluctuations for macroscopic observables of the random field Curie-Weiss model (i.e., standard Curie-Weiss model embedded in a site-dependent, i.i.d. random environment). We obtain path-space moderate deviation principles via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton–Jacobi equations. The moderate asymptotics depend crucially on the phase we consider and moreover, the space-time scale range for which fluctuations can be proven is restricted by the addition of the disorder.

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued “random-field” solutions to these SPDEs exist and are jointly Hölder continuous in space and time. We calculate the respective Hölder exponents, which extend the well-known results on the Hölder exponents of the solution to SHE on the unit interval. This shows that the “curse of dimensionality” of the SHE on $\mathbb{R} ^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov’s continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.

We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and Hénard [5] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total measure will converge to a point measure at its extinction time. This partially generalizes a result of Tribe [27] in the sense that our branching mechanism is more general.

We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with \(\binom{n} {2} + n= \tfrac 12n(n+1)\). integrals of the form $\int W_1^iW_2^j \circ{\operatorname {d}} W_2$, where $j=1,\ldots ,n$ and $i=0, \ldots , n-j$ for some fixed $n$. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions.

A sequence of chains exhibits (total variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon < 1/2$, the ratio $t_{\mathrm{mix} }^{(n)}(\epsilon )/t_{\mathrm{mix} }^{(n)}(1-\epsilon )$ tends to 1 as $n \to \infty $ (resp., the $\limsup $ of this ratio is bounded uniformly in $\epsilon $), where $t_{\mathrm{mix} }^{(n)}(\epsilon )$ is the $\epsilon $-total variation mixing time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the “product condition” $\mathrm{gap} (G_n) t_{\mathrm{mix} }^{(n)}(\epsilon ) \to \infty $ as $n \to \infty $, where $\mathrm{gap} (G_n)$ is the spectral gap of the LSRW on $G_n$ (a known necessary condition for pre-cutoff that is often sufficient for cutoff), yet this sequence does not exhibit pre-cutoff.

Recently, Chen and Saloff-Coste showed that total variation cutoff is equivalent for the sequences of continuous-time and lazy versions of some given sequence of chains. Surprisingly, we show that this is false when considering separation cutoff.

We also construct a sequence of bounded degree graphs $G_n=(V_{n},E_{n})$ that does not exhibit cutoff, for which a certain bounded perturbation of the edge weights leads to cutoff and increases the order of the mixing time by an optimal factor of $\Theta (\log |V_n|)$. Similarly, we also show that “lumping” states together may increase the order of the mixing time by an optimal factor of $\Theta (\log |V_n|)$. This gives a negative answer to a question asked by Aldous and Fill.

The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this paper to mix for $\frac{1} {2} \leq p \leq 1$ fixed, $n$ sufficiently large in between $\log _{1/p}(n)$ steps and $\log _{2/(1+p)}(n)$ steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. Monotonicity relations used in the bound also give after sufficient time the likelihood order, the asymptotic order from most likely to least likely permutation. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of black holes.

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.

We consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential over the two dimensional torus. We prove existence and uniqueness of solutions to an initial value problem for suitable initial data. Our construction is based on a change of unknown originally used in [13] and conserved quantities.

The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even in very simple cases.

Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha $-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer’s endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.

We show that, for general convolution approximations to a large class of log-correlated fields, including the 2d Gaussian free field, the critical chaos measures with derivative normalisation converge to a limiting measure $\mu '$. This limiting measure does not depend on the choice of approximation. Moreover, it is equal to the measure obtained using the Seneta–Heyde renormalisation at criticality, or using a white-noise approximation to the field.

We prove an exponential concentration bound for cover times of general graphs in terms of the Gaussian free field, extending the work of Ding, Lee, and Peres [8] and Ding [7]. The estimate is asymptotically sharp as the ratio of hitting time to cover time goes to zero.

The bounds are obtained by showing a stochastic domination in the generalized second Ray-Knight theorem, which was shown to imply exponential concentration of cover times by Ding in [7]. This stochastic domination result appeared earlier in a preprint of Lupu [22], but the connection to cover times was not mentioned.

Let $P_n^1,\dots , P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum _{\ell =1}^d P_n^\ell $. We show that if $\log ^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d} $ converges weakly to the circular law in probability as $n \to \infty $.

It is known that in an irreducible small Pólya urn process, the composition of the urn after suitable normalization converges in distribution to a normal distribution. We show that if the urn also is balanced, this normal convergence holds with convergence of all moments, thus giving asymptotics of (central) moments.

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.

Finally, a transfer principle “from-Poisson-to-Gaussian” is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.

The present paper addresses the following question: for a geometric random tree in $\mathbb{R} ^{2}$, how many semi-infinite branches cross the circle $\mathcal{C} _{r}$ centered at the origin and with a large radius $r$? We develop a method ensuring that the expectation of the number $\chi _{r}$ of these semi-infinite branches is $o(r)$. The result follows from the fact that, far from the origin, the distribution of the tree is close to that of an appropriate directed forest which lacks bi-infinite paths. In order to illustrate its robustness, the method is applied to three different models: the Radial Poisson Tree (RPT), the Euclidean First-Passage Percolation (FPP) Tree and the Directed Last-Passage Percolation (LPP) Tree. Moreover, using a coalescence time estimate for the directed forest approximating the RPT, we show that for the RPT $\chi _{r}$ is $o(r^{1-\eta })$, for any $0<\eta <1/4$, almost surely and in expectation.

Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta models. This integrability property encapsulates a preservation in distribution of ratios of partition functions which in turn implies the so called Burke property. We show that under some regularity assumptions, up to trivial modifications, there exist no other models possessing this property.

This article studies vertex reinforced random walks that are non-backtracking (denoted VRNBW), i.e. U-turns forbidden. With this last property and for a strong reinforcement, the emergence of a path may occur with positive probability. These walks are thus useful to model the path formation phenomenon, observed for example in ant colonies. This study is carried out in two steps. First, a large class of reinforced random walks is introduced and results on the asymptotic behavior of these processes are proved. Second, these results are applied to VRNBWs on complete graphs and for reinforced weights $W(k)=k^\alpha $, with $\alpha \ge 1$. It is proved that for $\alpha >1$ and $3\le m< \frac{3\alpha -1} {\alpha -1}$, the walk localizes on $m$ vertices with positive probability, each of these $m$ vertices being asymptotically equally visited. Moreover the localization on $m>\frac{3\alpha -1} {\alpha -1}$ vertices is a.s. impossible.

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim _{n \to \infty } r_n = \infty $ and $\limsup _{n \to \infty } r_n/n <1$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty } (dx) \otimes e^{-2\sqrt{\log b} ~y}dy$, where $\alpha > 0$ is a constant and $Z_{\infty }$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.

We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $\Xi $-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.

In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $\Xi $-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a. s. càdlàg paths with jumps at the times of large reproduction events. In the case of coming down from infinity, the construction on the lookdown space also allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a. s. càdlàg paths with additional jumps at the extinction times of parts of the population.

One dimensional excited random walk has been extensively studied for bounded, i.i.d. cookie environments. In this case, many important properties of the walk including transience or recurrence, positivity or non-positivity of the speed, and the limiting distribution of the position of the walker are all characterized by a single parameter $\delta $, the total expected drift per site. In the more general case of stationary ergodic environments, things are not so well understood. If all cookies are positive then the same threshold for transience vs. recurrence holds, even if the cookie stacks are unbounded. However, it is unknown if the threshold for transience vs. recurrence extends to the case when cookies may be negative (even for bounded stacks), and moreover there are simple counterexamples to show that the threshold for positivity of the speed does not. It is thus natural to study the behavior of the model in the case of Markovian environments, which are intermediate between the i.i.d. and stationary ergodic cases. We show here that many of the important results from the i.i.d. setting, including the thresholds for transience and positivity of the speed, as well as the limiting distribution of the position of the walker, extend to a large class of Markovian environments. No assumptions are made about the positivity of the cookies.

We prove that the flux function of the totally asymmetric simple exclusion process (TASEP) with site disorder exhibits a flat segment for sufficiently dilute disorder. For high dilution, we obtain an accurate description of the flux. The result is established under a decay assumption of the maximum current in finite boxes, which is implied in particular by a sufficiently slow power tail assumption on the disorder distribution near its minimum. To circumvent the absence of explicit invariant measures, we use an original renormalization procedure and some ideas inspired by homogenization.

We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is \[ \varepsilon _{n}(f):={\mathbb{E} }\Big (f\Big (\frac 1{\sqrt n}\sum _{i=1}^{n}Z_{i}\Big )\Big )-{\mathbb{E} }\big (f(G)\big )\rightarrow 0 \] where $Z_{i}$, $i\in \mathbb{N} $, are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\in{\mathbb {N}} $, on hand is needed. Essentially, one needs that the law of $Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin’s condition). This topic is also widely discussed in the literature. Our main contribution is to discuss convergence in distribution norms, that is to replace the test function $f$ by some derivative $\partial _{\alpha }f$ and to obtain upper bounds for $\varepsilon _{n}(\partial _{\alpha }f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.

We consider infinite Galton-Watson trees without leaves together with i.i.d. random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in [12], to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda $-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability that it returns to the true root state converges to $1$ as the number of leaves tends to infinity. We also derive quantitative bounds on the error of reconstruction. Our results answer a question of Gascuel and Steel [GS10] and have implications for ancestral sequence reconstruction in a classical evolutionary model of nucleotide insertion and deletion [TKF91].

We compute the exact decay rate of the hole probabilities for $\beta $-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory for both processes.

We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i \neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($\Xi $-)coalescent. This complements Möhle and Sagitov’s (2001) result for the haploid case and sharpens the profile of Möhle and Sagitov’s (2003) study of the diploid case, which focused on fixed couples, where each row of $V$ has at most one non-zero entry.

We apply the convergence result to several examples, in particular to two diploid variations of Schweinsberg’s (2003) model, leading to Beta-coalescents with two-fold and with four-fold mergers, respectively.

The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales has the decomposition similar to the classical case. The main ideas are to apply the property on uniform continuity of $S_G^\beta (0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.

We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles $\rho \in (0,1)$. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.

Consider a rooted infinite Galton–Watson tree with mean offspring number $m>1$, and a collection of i.i.d. positive random variables $\xi _e$ indexed by all the edges in the tree. We assign the resistance $m^d\,\xi _e$ to each edge $e$ at distance $d$ from the root. In this random electric network, we study the asymptotic behavior of the effective resistance and conductance between the root and the vertices at depth $n$. Our results generalize an existing work of Addario-Berry, Broutin and Lugosi on the binary tree to random branching networks.

We study the isoperimetric subgraphs of the giant component $\mathbf{{C}} _n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\mathbf{{C}} _n$ with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within $\mathbf{{C}} _n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\mathbf{{C}} _n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.

We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness ($\mathsf{UIHPQ} _p$ for short, with $p\in [0,1/2]$ measuring the skewness). They interpolate between Kesten’s tree corresponding to $p=0$ and the usual $\mathsf{UIHPQ} $ with a general boundary corresponding to $p=1/2$. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that the family $(\mathsf{UIHPQ} _p)_p$ approximates the Brownian half-planes $\mathsf{BHP} _\theta $, $\theta \geq 0$, recently introduced in [8]. For $p<1/2$, we give a description of the $\mathsf{UIHPQ} _p$ in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.

Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.

We establish two results concerning a class of geometric rough paths $\mathbf{X} $ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X} $ in $\alpha $-Hölder rough path topology for all $\alpha \in (0,1/2)$, which proves a conjecture of Friz–Victoir [13]. The second is a Hörmander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X} $, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.

We study Swendsen–Wang dynamics for the critical $q$-state Potts model on the square lattice. For $q=2,3,4$, where the phase transition is continuous, the mixing time $t_{\mathrm{mix} }$ is expected to obey a universal power-law independent of the boundary conditions. On the other hand, for large $q$, where the phase transition is discontinuous, the authors recently showed that $t_{\mathrm{mix} }$ is highly sensitive to boundary conditions: $t_{\mathrm{mix} } \geq \exp (cn)$ on an $n\times n$ box with periodic boundary, yet under free or monochromatic boundary conditions, $t_{\mathrm{mix} } \leq \exp (n^{o(1)})$.

In this work we classify this effect under boundary conditions that interpolate between these two (torus vs. free/monochromatic). Specifically, if one of the $q$ colors is red, mixed boundary conditions such as red-free-red-free on the 4 sides of the box induce $t_{\mathrm{mix} } \geq \exp (cn)$, yet Dobrushin boundary conditions such as red-red-free-free, as well as red-periodic-red-periodic, induce sub-exponential mixing.

Let $X_{\alpha }=\{X_{\alpha }(t),t\in{\cal T} \}$, $\alpha >0$, be an $\alpha $-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha }$ is a subgaussian process with respect to the metric \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\nonumber \] This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha $-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient Lévy processes that are not necessarily symmetric, or with kernels of the form \[ \widetilde{u} (x,y)= u(x,y)+f(y),\nonumber \] where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.

In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]}B_{t+u} \right \}$ for $t \geq 0$. The argmin process $\alpha $ is stationary, with invariant measure which is arcsine distributed. We prove that $(\alpha _t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot )$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed renewal process with an explicit path construction. We also give a path decomposition for Brownian motion at these extrema.

We study two-valued local sets, $\mathbb{A} _{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. Intuitively, $\mathbb{A} _{-a,b}$ is the (random) set of points connected to the boundary by a path on which the values of the GFF remain in $[-a,b]$. For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model.

Two-valued sets are the closure of the union of countably many SLE$_4$ type of loops, where each loop comes with a label equal to either $-a$ or $b$. One of the main results of this paper describes the connectivity properties of these loops. Roughly, we show that all the loops are disjoint if $a+b \geq 4\lambda $, and that their intersection graph is connected if $a + b < 4\lambda $. This also allows us to study the labels (the heights) of the loops. We prove that the labels of the loops are a function of the set $\mathbb{A} _{-a,b}$ if and only if $a\neq b$ and $2\lambda \leq a+b < 4\lambda $ and that the labels are independent given the set if and only if $a = b = 2\lambda $. We also show that the threshold for the level-set percolation in the 2D continuum GFF is $-2\lambda $.

Finally, we discuss the coupling of the labelled CLE$_4$ with the GFF. We characterise this coupling as a specific local set coupling, and show how to approximate these local sets. We further see how in these approximations the labels naturally encode distances to the boundary.

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon >0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, \[\exp \left ( - t^{ - \frac 1 { 1 + \frac 1 2 \gamma ^2 } - \varepsilon } \right ) \le p_t^\gamma (x, y) \le \exp \left ( - t^{- \frac 1 { 1 + \frac 1 2 \gamma ^2 } + \varepsilon } \right ) ,\] for $\gamma <1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field.

We prove sharpness of the phase transition for the random-cluster model with $q \geq 1$ on graphs of the form $\mathcal{S} := \mathcal{G} \times S$, where $\mathcal{G} $ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q \geq 1$, there exists some parameter $p_c = p_c(\mathcal{S} , q)$, below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.

In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and the Laplace exponent of the corresponding subordinator only.

In this article, we study weighted particle representations for a class of stochastic partial differential equations (SPDE) with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations. The locations are given by independent, stationary reflecting diffusions in a bounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise $W$ which is the stochastic input for the SPDE. The weights interact through $V$, the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on the boundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure $V$ is the unique solution for each of the nonlinear stochastic partial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14, 15].

We consider a particular type of $\sqrt{8/3} $-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $\eta $ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta ([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3} $-quantum boundary length process for $\eta $.

Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we are interested in the height and contour processes encoding a general CMJ tree.

We show that the one-dimensional distribution of the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. As an application of this result, when edges of the tree are “short” we show that, asymptotically, (1) the height process is obtained by stretching by a constant factor the height process of the associated genealogical Galton–Watson tree, (2) the contour process is obtained from the height process by a constant time change and (3) the CMJ trees converge in the sense of finite-dimensional distributions.

We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $\mathbb{Z} ^d$ ($d\geq 2$) with zero Dirichlet condition. We assume that the conductances $w$ are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If $\gamma =\sup \{ q\geq 0\colon \mathbb{E} [w^{-q}]<\infty \}<1/4$, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold $\gamma _{\rm c} = 1/4$ is sharp. Indeed, other recent results imply that for $\gamma >1/4$ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

We give necessary and sufficient conditions for a $d$–dimensional Lévy process $({\bf X}_t)_{t\ge 0}$ to be in the matrix normalised Feller (stochastic compactness) classes $FC$ and $FC_0$ as $t\downarrow 0$. This extends earlier results of the authors concerning convergence of a Lévy process in $\Bbb{R} ^d$ to normality, as the time parameter tends to 0. It also generalises and transfers to the Lévy case classical results of Feller and Griffin concerning real- and vector-valued random walks. The process $({\bf X}_t)$ and its quadratic variation matrix together constitute a matrix-valued Lévy process, and, in a further extension, we show that the condition derived for the process itself also guarantees the stochastic compactness of the combined matrix-valued process. This opens the way to further investigations regarding self-normalised processes.

We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N} $, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are based on the flow of the resolvents of matrices with a random diagonal component under Dyson Brownian motion, for which we establish submicroscopic stability results for short times. These results go beyond norm-based continuity arguments for Dyson Brownian motion and complement the existing analysis after the local equilibration time.

It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors [10], where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of [6], who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.

In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate that an individual of type $x$ experiences due to the presence of an individual or type $y$. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may expresses a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed in [9, 11].

We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).

We introduce an efficient numerical scheme for continuous time Dynkin games under model uncertainty. We use the Skorokhod embedding in order to construct recombining tree approximations. This technique allows us to determine convergence rates and to construct numerically optimal stopping strategies. We apply our method to several examples of game options.

In this work we analyse the solution to the recurrence equation \[ M_\Psi (z+1)=\frac{-z} {\Psi (-z)}M_\Psi (z), \quad M_\Psi (1)=1, \] defined on a subset of the imaginary line and where $-\Psi $ is any continuous negative definite function. Using the analytic Wiener-Hopf method we solve this equation as a product of functions that extend the gamma function and are in bijection with the Bernstein functions. We call these functions Bernstein-gamma functions. We establish universal Stirling type asymptotic in terms of the constituting Bernstein function. This allows the full understanding of the decay of $\vert M_\Psi (z)\vert $ along imaginary lines and an access to quantities important for many studies in probability and analysis.

This functional equation is a central object in several recent studies ranging from analysis and spectral theory to probability theory. As an application of the results above, we study from a global perspective the exponential functionals of Lévy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Lévy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions and bounds. We furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. We deliver intertwining relation between members of the class of positive self-similar semigroups.

Sequences of certain finite graphs - special types of antitrees - are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\mathrm{Sine} }_1$ process. The Anderson model on the graph is a random matrix being the sum of the adjacency matrix and a random diagonal matrix with independent identically distributed entries along the diagonal. The strength of the randomness stays fixed, there is no re-scaling with matrix size. These considered random matrices giving GOE statistics can also be viewed as random Schrödinger operators $\mathcal{P} \Delta +\mathcal{V} $ on thin finite boxes in $\mathbb{Z} ^3$ where the Laplacian $\Delta $ is deformed by a projection $\mathcal{P} $ commuting with $\Delta $.

We study random band matrices within the framework of traffic probability. As a starting point, we revisit the familiar case of permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We further prove general Markov-type concentration inequalities for the joint traffic distribution. We then extend our analysis to random band matrices and investigate the extent to which the joint traffic distribution of independent copies of these matrices deviates from the Wigner case.

We construct the Dirichlet form associated with the dynamical $\Phi ^4_3$ model obtained in [23, 7] and [37]. This Dirichlet form on cylinder functions is identified as a classical gradient bilinear form. As a consequence, this classical gradient bilinear form is closable and then by a well-known result its closure is also a quasi-regular Dirichlet form, which means that there exists another (Markov) diffusion process, which also admits the $\Phi ^4_3$ field measure as an invariant (even symmetrizing) measure.

A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele’s algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0< q <1$. The classic Berele algorithm corresponds to letting the parameter $q\to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type identities for a $q$-deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$-Hermite polynomials. Finally, we show that when both the original and the $q$-modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.

We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq} = \overline{H} _{qp} = \pm i$, that are either independently distributed or exhibit global correlations imposed by the condition $\sum _{q} H_{pq} = 0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein’s method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\mathrm{sgn} (\varphi )$ where $\varphi $ is a discrete Gaussian free field.

In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.

The antiferromagnetic $q$-state Potts model is perhaps the most canonical model for which the uniqueness threshold on the tree is not yet understood, largely because of the absence of monotonicities. Jonasson established the uniqueness threshold in the zero-temperature case, which corresponds to the $q$-colourings model. In the permissive case (where the temperature is positive), the Potts model has an extra parameter $\beta \in (0,1)$, which makes the task of analysing the uniqueness threshold even harder and much less is known.

In this paper, we focus on the case $q=3$ and give a detailed analysis of the Potts model on the tree by refining Jonasson’s approach. In particular, we establish the uniqueness threshold on the $d$-ary tree for all values of $d\geq 2$. When $d\geq 3$, we show that the 3-state antiferromagnetic Potts model has uniqueness for all $\beta \geq 1-3/(d+1)$. The case $d=2$ is critical since it relates to the 3-colourings model on the binary tree ($\beta =0$), which has non-uniqueness. Nevertheless, we show that the Potts model has uniqueness for all $\beta \in (0,1)$ on the binary tree. Both of these results are tight since it is known that uniqueness does not hold in the complementary regime.

Our proof technique gives for general $q>3$ an analytical condition for proving uniqueness based on the two-step recursion on the tree, which we conjecture to be sufficient to establish the uniqueness threshold for all non-critical cases ($q\neq d+1$).

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in $\mathbb{Z} ^d$. This endeavor solves a problem of Barlow and Taylor (1991).

We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i.e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leq \delta _0,\psi _1(x_1)<x_2<\psi _ 2(x_1)\}$, with $\psi _1(0)=\psi _2(0)=0$, $\psi _1'(0)=\psi _2'(0)=0$.

Given a vector field $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $ i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin.

Our proof uses a new scaling result and a coupling argument.

We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval $[0,\lambda ]$ as $\lambda \to \infty $. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the $\mathrm{Sine} _\beta $ process.

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary $ L^2$ computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, we use convergence of conditioned Galton-Watson trees towards stable Lévy trees, which provides less precise results but leads us to conjecture a different phase transition value between “global” and “local” regimes. We also recover results first given by Janson (2003 and 2016) in the Brownian case and give a generalization to the stable case.

We study the frog model on $\mathbb{Z} ^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

We introduce the Incipient Infinite Cluster ($\mathsf{IIC} $) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation ($\mathsf{UIHPT} $), which is the local limit of large random triangulations with a boundary. The $\mathsf{IIC} $ is defined from the $\mathsf{UIHPT} $ by conditioning the open percolation cluster of the origin to be infinite. We prove that the $\mathsf{IIC} $ can be obtained by adding within the $\mathsf{UIHPT} $ an infinite triangulation with a boundary whose distribution is explicit.

We consider $d$ independent walkers on $\mathbb{Z} $, $m$ of them performing simple symmetric random walk and $r= d-m$ of them performing recurrent RWRE (Sinai walk), in $I$ independent random environments. We show that the product is recurrent, almost surely, if and only if $m\leq 1$ or $m=d=2$. In the transient case with $r\geq 1$, we prove that the walkers meet infinitely often, almost surely, if and only if $m=2$ and $r \geq I= 1$. In particular, while $I$ does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.

The exponential stability and the concentration properties of a class of extended Kalman-Bucy filters are analyzed. New estimation concentration inequalities around partially observed signals are derived in terms of the stability properties of the filters. These non asymptotic exponential inequalities allow to design confidence interval type estimates in terms of the filter forgetting properties with respect to erroneous initial conditions. For uniformly stable and fully observable signals, we also provide explicit non-asymptotic estimates for the exponential forgetting rate of the filters and the associated stochastic Riccati equations w.r.t. Frobenius norms. These non asymptotic exponential concentration and quantitative stability estimates seem to be the first results of this type for this class of nonlinear filters. Our techniques combine $\chi $-square concentration inequalities and Laplace estimates with spectral and random matrices theory, and the non asymptotic stability theory of quadratic type stochastic processes.

Under correlation-type conditions, we derive upper bounds of order $\frac{1} {\sqrt{n} }$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables.

To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vallée on the asymptotic normality of subword counts in random texts generated by a Markovian source.

We consider the compact space of pairs of nested partitions of $\mathbb{N} $, where by analogy with models used in molecular evolution, we call “gene partition” the finer partition and “species partition” the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $\Lambda $-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo $\Lambda $-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. $\Lambda $-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure $\nu _s$ on $(0,1]\times{\mathcal M} _1 ([0,1])$ providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

We consider continuous-time Markov chains which display a family of wells at the same depth. We provide sufficient conditions which entail the convergence of the finite-dimensional distributions of the order parameter to the ones of a finite state Markov chain. We also show that the state of the process can be represented as a time-dependent convex combination of metastable states, each of which is supported on one well.

We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in $1+1$ dimensions.

We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results.

We consider a spectrally positive Lévy process $X$ that does not drift to $+\infty $, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf _{[0,t]} X$ and we let $\gamma $ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma $ and the post-$\gamma $ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of Lévy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

We consider the model of i.i.d. first passage percolation on $\mathbb{Z} ^d$, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution $G$ on $ [0,+\infty ]$ (including $+\infty $). Whereas the time constant is associated to the study of $1$-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of $(d-1)$-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that $G(\{+\infty \} ) < p_c(d)$ (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution $G$.

Given a non-negative random variable $W$ and $\theta >0$, let the generalized Dickman transformation map the distribution of $W$ to that of \[ W^*=_d U^{1/\theta }(W+1), \] where $U \sim{\cal U} [0,1]$, a uniformly distributed variable on the unit interval, independent of $W$, and where $=_d$ denotes equality in distribution. It is well known that $W^*$ and $W$ are equal in distribution if and only if $W$ has the generalized Dickman distribution ${\cal D}_\theta $. We demonstrate that the Wasserstein distance $d_1$ between $W$, a non-negative random variable with finite mean, and $D_\theta $ having distribution ${\cal D}_\theta $ obeys the inequality \[ d_1(W,D_\theta ) \le (1+\theta )d_1(W,W^*). \] The specialization of this bound to the case $\theta =1$ and coupling constructions yield \[ d_1(W_{n,1},D_1) \le \frac{8\log (n/2)+10} {n} \quad \mbox{for all $n \ge 1$, where for $m \ge 1$} \quad W_{n,m}=\frac{1} {n}C_{n,m}-1, \] and $C_{n,m}$ is the number of comparisons made by the Quickselect algorithm to find the $m^{th}$ smallest element of a list of $n$ distinct numbers. A similar bound holds for $W_{n,m}$ for $m \ge 2$, and together recover and quantify the results of [12] that show distributional convergence of $W_{n,m}$ to the standard Dickman distribution ${\cal D}_1$ in the asymptotic regime $m=o(n)$. By comparison to an exact expression for the expected running time $E[C_{n,m}]$, lower bounds are provided that show the rate is not improvable for $m \not = 2$.

We consider different limit theorems for additive and multiplicative free Lévy processes. The main results are concerned with positive and unitary multiplicative free Lévy processes at small times, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large times to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean Lévy processes at small times, obtaining log Boolean stable laws in the limit.

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche [7] found the critical value and studied the subcritical and supercritical regimes. Recently Foxall [4] has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $\sqrt{N} $. Here we improve that result by proving the convergence of $i_N(t)=I(\sqrt{N} t)/\sqrt{N} $ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $\sqrt{N} $ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $\tau _N$ is the extinction time of $i_N(t)$ (on the $\sqrt{N} $ time scale) then $\tau _N$ has a limit.

We provide another construction of the natural parametrization of SLE$_\kappa $ [9, 8] for $\kappa < 4$. We construct it as the expectation of the quantum time [14], which is a random measure carried by SLE in an ambient Gaussian free field. This quantum time was built as the push forward on the SLE curve of the Liouville boundary measure, which is a natural field-dependent measure supported on the boundary of the domain. We moreover show that the quantum time can be reconstructed as a chaos on any measure on the trace of SLE with the right Markovian covariance property. This provides another proof that the natural parametrization is characterized by its Markovian covariance property.

We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $\mathcal{P} \subseteq \mathbb{R} ^d - \mathbb{B} (0,r)$. Around each point in $\mathcal{P} $, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x \in \mathcal{P} $, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated points in $\mathcal{P} $ approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $\mathbb{Z} ^d$, $d \geq 3$, below a level $\alpha $ disconnects the discrete blow-up of a compact set $A$ from the boundary of the discrete blow-up of a box that contains $A$, when the level set of the Gaussian free field above $\alpha $ is in a strongly percolative regime. These bounds substantially strengthen the results of [21], where $A$ was a box and the convexity of $A$ played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of $A$ is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work [15] of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of $A$, which can be understood as entropic repulsion.

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski in [11], and that depend on the inhomogeneities.

A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by an infinite urn scheme, where same values are repeated at several random locations. The random sup-measure reflects the long-range dependence nature of the original process, and in particular characterizes how locations of extremes appear as long-range clusters represented by random closed sets. A limit theorem for the corresponding point-process convergence is established.

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R} ^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

For the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta _0$, we study its asymptotic behavior as $x\to 0$. In $d=3$, we find a normalization $\psi (x)=((2\pi ^2)^{-1} \log (1/|x|))^{1/2}$ such that $(L_t^x-(2\pi |x|)^{-1})/\psi (x)$ converges in distribution to standard normal as $x\to 0$. In $d=2$, we show that $L_t^x-\pi ^{-1} \log (1/|x|)$ converges a.s. as $x\to 0$. We also consider general initial conditions and get some renormalization results. The behavior of the local time allows us to derive a second order term in the asymptotic behavior of a related semilinear elliptic equation.

For each $n$, let $A_n=(\sigma _{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $\mu _n^Y$ of the rescaled entry-wise product \[ Y_n = \left (\frac 1{\sqrt{n} } \sigma _{ij}X_{ij}\right ). \] For our main result we provide a deterministic sequence of probability measures $\mu _n$, each described by a family of Master Equations, such that the difference $\mu ^Y_n - \mu _n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma _{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger–Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

We study the images of the complex Ginibre eigenvalues under the power maps $\pi _M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, \[ \mathrm{Gin} (N)^M \stackrel{d} {=} \bigcup _{k=1}^M \mathrm{Gin} (N,M,k), \] where the so-called Power-Ginibre distributions $\mathrm{Gin} (N,M,k)$ form $M$ independent determinantal point processes. The decomposition can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains’ superposition theorem for the CUE (see [21]) and Kostlan’s independence of radii (see [17]) to a wider class of point processes. Our proof technique also allows us to recover two results by Edelman and La Croix [12] for the GUE.

Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of their smooth linear statistics to independent gaussian variables, coherent with the link between the complex Ginibre Ensemble and the Gaussian Free Field [22].

Finally, some partial results about two-dimensional beta ensembles with radial symmetry and even parameter $\beta $ are discussed, replacing independence by conditional independence.

A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional Itô stochastic differential equation, and the fast component is a finite state pure jump process. Previous works have considered settings where the coupling between the components is weak in a certain sense. In the current work we study a fully coupled system in which the drift and diffusion coefficient of the slow component and the jump intensity function and jump distribution of the fast process depend on the states of both components. In addition, the diffusion can be degenerate. Our proofs use certain stochastic control representations for expectations of exponential functionals of finite dimensional Brownian motions and Poisson random measures together with weak convergence arguments. A key challenge is in the proof of the large deviation lower bound where, due to the interplay between the degeneracy of the diffusion and the full dependence of the coefficients on the two components, the associated local rate function has poor regularity properties.

We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [15], by proving local estimates for the density, existence for the density with summable drift, and by improving the regularity of the density.

We study one-dimensional nearest neighbour random walk in site-dependent random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $\beta $-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

A cyclic urn is an urn model for balls of types $0,\ldots ,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector.

In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all $7\le m\le 12$. For $m\ge 13$ we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension $m-1$ only when $6$ does not divide $m$. For $m$ being a multiple of $6$ the fluctuations are supported by a two-dimensional subspace.

It is known from the work of Baik, Deift and Johansson [3] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutations with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting descent process of a large class of random permutations is stationary, one-dependent and determinantal.

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in ${\mathbb{R} }^{n}$.

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called ‘chronological tree’. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of ‘splitting tree’ studied in [Lam10], where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive Lévy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees. The genealogical trees associated to splitting trees are the celebrated Lévy trees in the subcritical case and they will be analyzed in the supercritical case in forthcoming work.

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete–time approximations of general martingales.

We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi $ and $\zeta $ do not satisfy any regularity assumption and with general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi $ is right upper-semicontinuous and $\zeta $ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\boldsymbol{\mathcal {E}} ^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of “an extension” of the previous non-linear game problem over a larger set of “stopping strategies” than the set of stopping times. This characterization is then used to establish a comparison result and a priori estimates with universal constants.

The Aldous gossip process represents the dissemination of information in geographical space as a process of locally deterministic spread, augmented by random long range transmissions. Starting from a single initially informed individual, the proportion of individuals informed follows an almost deterministic path, but for a random time shift, caused by the stochastic behaviour in the very early stages of development. In this paper, it is shown that, even with the extra information available after a substantial development time, this broad description remains accurate to first order. However, the precision of the prediction is now much greater, and the random time shift is shown to have an approximately normal distribution, with mean and variance that can be computed from the current state of the process.

We propose in this paper a construction of a diffusion process on the space $\mathcal P_2(\mathbb R)$ of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. [12]) and consists of the limit as $N$ tends to $+\infty $ of a system of $N$ coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on $\mathcal P_2(\mathbb R)$ constructed by von Renesse and Sturm in [22]. We obtain that process by the construction of a system of particles having short-range interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov’s theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Itô-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multidimensional Itô-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

We consider a model of language development, known as the naming game, in which agents invent, share and then select descriptive words for a single object, in such a way as to promote local consensus. When formulated on a finite and connected graph, a global consensus eventually emerges in which all agents use a common unique word. Previous numerical studies of the model on the complete graph with $n$ agents suggest that when no words initially exist, the time to consensus is of order $n^{1/2}$, assuming each agent speaks at a constant rate. We show rigorously that the time to consensus is at least $n^{1/2-o(1)}$, and that it is at most constant times $\log n$ when only two words remain. In order to do so we develop some useful estimates for semimartingales with bounded jumps.

Branching Brownian motion (BBM) is a convenient representative of the class of log-correlated random fields. Motivated by the conjectured criticality of the log-correlated fields, we take the viewpoint of statistical physics on the BBM: We consider the partition function of the field of energies given by the “positions” of the particles of the complex-valued BBM. In such a complex BBM energy model, we allow for arbitrary correlations between the real and imaginary parts of the energies. We identify the fluctuations of the partition function. As a consequence, we get the full phase diagram of the log-partition function. It turns out that the phase diagram is the same as for the field of independent energies, i.e., Derrida’s random energy model (REM). Yet, the fluctuations are different from those of the REM in all phases. All results are shown for any correlation between the real and imaginary parts of the random energy.

We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.

We extend a functional limit theorem for symmetric $U$-statistics [Miller and Sen, 1972] to asymmetric $U$-statistics, and use this to show some renewal theory results for asymmetric $U$-statistics.

Some applications are given.

We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on $\mathbb{Z} ^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+\alpha )}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<\alpha <1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $\alpha \geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.

We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf _{n \to \infty } L_n/D_n =2$ and $\limsup _{n \to \infty } L_n /D_n = \pi $, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.

We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.