In the classical balls-and-bins model, $m$ balls are allocated into $n$ bins one by one uniformly at random. In this note, we consider the $d$-thinning variant of this model, in which the process is regulated in an on-line fashion as follows. For each ball, after a random bin has been selected, an overseer may decide, based on all previous history, whether to accept this bin or not. However, one of every $d$ consecutive suggested bins must be accepted. The maximum load of this setting is the number of balls in the most loaded bin. We show that after $\Theta (n)$ balls have been allocated, the least maximum load achievable with high probability is $(d+o(1))\sqrt [d]{\frac{d\log n} {\log \log n}}$. This should be compared with the related $d$-choice setting, in which the optimal maximum load achievable with high probability is $\frac{\log \log n} {\log d}+O(1)$.

We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p>p_{c}$, whose existence has been recently proved in [3]. We extend adequately the definition for $p=p_{c}$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p_{c}$ exists, it has to vanish.

In this paper, under the assumption of Hölder continuous coefficients, we prove the strong Feller property and continuous dependence on initial data for the solution to one-dimensional stochastic differential equations whose proof are based on the technique of local time, coupling method and Girsanov’s transform.

Let $(X_{1},X_{2},...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_{i}\geq 0$ and $\sum _{i\geq 1} X_{i}=1$, and let $(\varepsilon _{1}, \varepsilon _{2},...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum _{i\geq 1} \varepsilon _{i} X_{i}$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in (0,1)$, what can we infer about the random partition $(X_{1}, X_{2},...)$? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $\nicefrac {1}{2}$.

If the step distribution in a renewal process has finite mean and regularly varying tail with index $-\alpha $, $1<\alpha <2$, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we show that, without making any additional assumptions, it is possible to give, in all cases except for $\alpha =3/2$, the exact asymptotic behaviour of the next term. In the case $\alpha =3/2$ the result is exact to within a slowly varying correction. Similar results are shown to hold in the random walk case.

The goal of this note is to show how recent results on the theory of quasi-stationary distributions allow us to deduce general criteria for the geometric convergence of normalized unbounded semigroups.

We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if \[ \lim _{x \to 0} \frac{\log \log 1/\mathbb {P} (W \leq x)} {\log \log 1/x} = \gamma > 1, \] then the Laplace transform of $Y$ satisfies \[ \lim _{t \to \infty } \frac{\log \log 1/\mathbb {E}e^{-t Y}} {\log \log t} = \gamma . \] This implies the same estimate for $\mathbb{P} (Y \leq x)$ for small $x > 0$. As an application of the method, we prove a similar result for a variable arising as a total mass of a $\star $-scale invariant Gaussian multiplicative chaos measure.

As a generalization of scale functions of spectrally negative Lévy processes, we define generalized scale functions of general standard processes with no positive jumps. For this purpose, we utilize the excursion theory. Using the generalized scale functions, we study Laplace transforms of hitting times, potential measures and duality.

We consider the transition semigroup $P_{t}$ of the $\Phi ^{4}_{2}$ stochastic quantisation on the torus $\mathbb {T}^{2}$ and prove the following new estimate (Theorem 3.10) \[ |DP_{t} \varphi (x)\cdot h|\le c\,t^{-\beta }\,|h|_{C^{-s}}\|\varphi \|_{0}\,(1+|x|_{C^{- \alpha }})^{\gamma }, \] for some $ \alpha ,\beta ,\gamma ,s$ positive. Thanks to this estimate, we show that cylindrical functions are a core for the corresponding Kolmogorov equation. Some consequences of this fact are discussed in a final remark.

Let $n,k\geq 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_{k},C_{k}>0$ depending only on $k$ such that the smallest singular value of $G^{k}$ satisfies \[ c_{k}\,t\leq{\mathbb {P}} \big \{s_{\min }(G^{k})\leq t^{k}\,n^{-1/2}\big \}\leq C_{k}\,t,\quad t\in (0,1], \] and, furthermore, \[ c_{k}/t\leq{\mathbb {P}} \big \{\|G^{-k}\|_{HS}\geq t^{k}\,n^{1/2}\big \}\leq C_{k}/t,\quad t\in [1,\infty ), \] where $\|\cdot \|_{HS}$ denotes the Hilbert–Schmidt norm.

We study the minimization of the expected costs under stochastic constraint at the terminal time. The first and the main result says that for a power type of costs, the value function is the minimal positive solution of a second order semi-linear ordinary differential equation (ODE). Moreover, we establish the optimal control. In the second example we show that the case of exponential costs leads to a trivial optimal control.

We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor $ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb{D} ([0, +\infty ), \mathcal{C} _{0}(\mathbb{R} ))$, as $N\to \infty $. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.

The extra-clustering model for the group formation process of social animals was introduced by Durand, Blum and François. The model uses the relatedness of the animals, which is described by phylogenetic trees. If these trees are drawn from the Yule-Harding model, it was analyzed in recent work. Here, we analyze it for the uniform model, which is the other widely-studied model on phylogenetics trees. More precisely, we derive moments and limit laws for the number of groups, the number of groups of fixed size and the largest group size. Our results show that, independent of the probability of extra-clustering, there is on average only a finite number of groups, one of which is very large whereas all others are small. This behavior considerably differs from the Yule-Harding case, where the finiteness of the number of groups is dependent on the extra-clustering probability.

We establish bounds on the probability that two different agents, who share an initial opinion expressed as a probability distribution on an abstract probability space, given two different sources of information, may come to radically different opinions regarding the conditional probability of the same event.

We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains. Our proof uses the same martingale decomposition as [2] but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.

When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Itô diffusions, using as our main tool a refinement of Strassen’s result due to Lerche (1986).

We construct a noise stable sequence of transitive, monotone increasing Boolean functions $f_{n}: \{-1,1\}^{k_{n}} \longrightarrow \{-1,1\}$ which admit many pivotals with high probability. We show that such a sequence is volatile as well, and thus it is also an example of a volatile and noise stable sequence of transitive, monotone functions.

Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance.

In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.

The rotor walk is a deterministic analogue of the simple random walk. For any given graph, we construct a rotor configuration for which the escape rate of the corresponding rotor walk is equal to the escape rate of the simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).

In [7] it was proved that, given a distribution $\mu $ with zero mean and finite second moment, there exists a simply connected domain $\Omega $ such that if $Z_{t}$ is a standard planar Brownian motion, then $\mathcal{R} e(Z_{\tau _{\Omega }})$ has the distribution $\mu $, where $\tau _{\Omega }$ denotes the exit time of $Z_{t}$ from $\Omega $. In this note, we extend this method to prove that if $\mu $ has a finite $p$-th moment then the first exit time $\tau _{\Omega }$ from $\Omega $ has a finite moment of order $\frac{p} {2}$. We also prove a uniqueness principle for this construction, and use it to give several examples.

We prove a sharp inequality for the ratio of the maximal functions of nonnegative martingales and their differential subordinates. An application in the theory of weighted inequalities is given.

We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue particles only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passage times and red rate-$\lambda $, a phase transition occurs for the probability red escapes to infinity on $\mathbb{Z} ^{d}$, $d$-ary trees, and the ladder graph $\mathbb{Z} \times \{0,1\}$. The result on the tree was known, but we provide a new, simpler calculation of the critical value, and observe that it is a lower bound for a variety of graphs. We conclude by showing that red can be stochastically slower than blue, but still escape with positive probability for large enough $d$ on oriented $\mathbb{Z} ^{d}$ with passage times that resemble Bernoulli bond percolation.

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V$-geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V$-geometrical ergodicity.

A little over 25 years ago Pemantle [6] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $\lambda _{1}$ and $\lambda _{2}$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is $(n,a_{1},\ldots , a_{k})$ with $\max _{i} a_{i} \le Cn^{1-\delta }$ and $\log (a_{1} \cdots a_{k})/\log n \to b$ as $n\to \infty $. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n} $ where $c=(k-b)/2$. This supports Pemantle’s claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

We present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.

We compute the Wassertein-1 (or Kantorovitch-Rubinstein) distance between a random walk in $\mathbf{R} ^{d}$ and the Brownian motion. The proof is based on a new estimate of the modulus of continuity of the solution of the Stein’s equation. As an application, we can evaluate the rate of convergence towards the local time at 0 of the Brownian motion and to a Brownian bridge.

In 2018, Kahle and Stump raised the following problem: identify sequences of finite Coxeter groups $W_{n}$ for which the two-sided descent statistics on a uniform random element of $W_{n}$ is asymptotically normal. Recently, Brück and Röttger provided an almost-complete answer, assuming some regularity condition on the sequence $W_{n}$. In this note, we provide a shorter proof of their result, which does not require any regularity condition. The main new proof ingredient is the use of the second Wasserstein distance on probability distributions, based on the work of Mallows (Ann. Math. Statist., 1972).

We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a “best” extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the sense of Chen-Fukushima. This extension, however, does not correspond to the bridging extension, which was introduced by Boscain-Prandi, when they previously considered self-adjoint extensions of the Laplace-Beltrami operator on the Riemannian part for these surfaces. We clarify that several of the extensions they considered induce diffusions that are carried by the Martin compactification at the singularity, which is much larger than the (one-point) metric completion. In the case when the singularity is more than one-point, a complete classification of diffusions extending Brownian motion would be unwieldy. Nonetheless, we again describe a “best” extension which respects the isometry group, and in this case, this diffusion corresponds to the bridging extension. A prominent role is played by Bessel processes (of every real dimension) and the classical theory of one-dimensional diffusions and their boundary conditions.

The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation lies in the identity between solutions and generators given by the representation theorem. Compared with existing methods, our approach seems to be a feasible unified method for different frameworks and be more applicable to general settings. This can also be regarded as a new application of such representation theorem.

The proof of the second convergence of Theorem 2.1 provided in the original paper relies on properties of the so called Q-process, which is not, in general, defined on the whole state space. Hence the inequality is only proved for some restricted sets of functions and measures. This weakness can be easily repaired and we provide here a proof of the inequality that applies in full generality.

We identify simple mechanisms that prevent a strong form of disorder chaos for edge overlap of the Ising spin glass model on $\mathbb{Z} ^{d}$. This was first shown by Chatterjee in the case of Gaussian couplings. We present three proofs of the theorem for general couplings with continuous distribution based on the presence in the coupling realization of stabilizing features of positive density.

We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.

We consider the standard model of i.i.d. first passage percolation on $\mathbb {Z}^{d}$ given a distribution $G$ on $\mathbb {R}_{+}$. We consider a cube oriented in the direction $\overrightarrow {v}$ whose sides have length $n$. We study the maximal flow from the top half to the bottom half of the boundary of this cube. We already know that the maximal flow renormalized by $n^{d-1}$ converges towards the flow constant $\nu _{G}(\overrightarrow {v})$. We prove here that the map $p\mapsto \nu _{p\delta _{1}+(1-p)\delta _{0}}$ is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$ where $p_{c}(d)$ denotes the critical parameter for i.i.d. bond percolation on $\mathbb {Z}^{d}$. For $p>p_{c}(d)$, we know that there exists almost surely a unique infinite open cluster $\mathcal {C}_{p}$ [8]. We are interested in the regularity properties in $p$ of the anchored isoperimetric profile of the infinite cluster $\mathcal {C}_{p}$. For $d\geq 2$, using the result on the regularity of the flow constant, we prove here that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$.

This paper deals with the rate of convergence in $1$-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached $0$ towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including probability measures with compact support, the Wasserstein distance decays asymptotically as $1/t$. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infinity.

We consider random trigonometric polynomials of the form \[ f_{n}(t):=\frac{1} {\sqrt{n} } \sum _{k=1}^{n}a_{k} \cos (k t)+b_{k} \sin (k t), \] where $(a_{k})_{k\geq 1}$ and $(b_{k})_{k\geq 1}$ are two independent stationary Gaussian processes with the same correlation function $\rho : k \mapsto \cos (k\alpha )$, with $\alpha \geq 0$. We show that the asymptotics of the expected number of real zeros differ from the universal one $\frac{2} {\sqrt{3} }$, holding in the case of independent or weakly dependent coefficients. More precisely, for all $\varepsilon >0$, for all $\ell \in (\sqrt{2} ,2]$, there exists $\alpha \geq 0$ and $n\geq 1$ large enough such that \[ \left |\frac{\mathbb {E} \left [\mathcal {N}(f_{n},[0,2\pi ])\right ]} {n}-\ell \right |\leq \varepsilon , \] where $\mathcal{N} (f_{n},[0,2\pi ])$ denotes the number of real zeros of the function $f_{n}$ in the interval $[0,2\pi ]$. Therefore, this result provides the first example where the expected number of real zeros does not converge as $n$ goes to infinity by exhibiting a whole range of possible subsequential limits ranging from $\sqrt{2} $ to 2.

The branching Brownian sausage in $\mathbb{R} ^{d}$ was defined in [4] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R} ^{d}$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated branching Brownian sausage (BBM-sausage) with radius exponentially shrinking in time. Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.

We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with $\alpha $-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths.

This paper is devoted to a direct martingale approach for Pólya urn models asymptotic behaviour. A Pólya process is said to be small when the ratio of its replacement matrix eigenvalues is less than or equal to $1/2$, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urn processes. We also provide new almost sure properties for small urn processes.

In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of $n$ consecutive random variables of a stationary Markov chain is linear in $n$. The paper also addresses the central limit theorem problem and is listing several open questions.

This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson–Voronoi cells, as well as power-weighted edge lengths in the random geometric, $k$-nearest neighbor and relative neighborhood graph.

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset \mathbb {C}$ whose exit time embeds a given probability distribution $\mu $ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $\mu $ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb {U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb {U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.

Let $T_{\lambda }$ be a Galton–Watson tree with Poisson($\lambda $) offspring, and let $A$ be a tree property. In this paper, we are concerned with the regularity of the function $\mathbb {P}_{\lambda }(A)\coloneqq \mathbb {P}(T_{\lambda }\models A)$. We show that if a property $A$ can be uniformly approximated by a sequence of properties $\{A_{k}\}$, depending only on the first $k$ vertices in the breadth first exploration of the tree, with a bound in probability of $\mathbb {P}_{\lambda }(A\triangle A_{k}) \le Ce^{-ck}$ over an interval $I = (\lambda _{0}, \lambda _{1})$, then $\mathbb {P}_{\lambda }(A)$ is real analytic in $\lambda $ for $\lambda \in I$. We also present some applications of our results, particularly to properties that are not expressible in first order logic on trees.

This article concerns the construction of a continuous branching process in a random, time-independent environment, on finite volume. The backbone of this study is the convergence of discrete approximations of the parabolic Anderson model (PAM) on a box with Dirichlet boundary conditions. This is a companion paper to [9].

In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting compound distributions. Although this is a classical textbook result in risk theory, our proof exhibits a concrete coupling between the compound distributions being compared, using the representation of one-period discrete martingale laws as a mixture of the corresponding extremal measures.

In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges recently obtained in [2] to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.

Consider critical site percolation on $\mathbb{Z} ^{d}$ with $d \geq 2$. We prove a lower bound of order $n^{- d^{2}}$ for point-to-point connection probabilities, where $n$ is the distance between the points.

Most of the work in our proof concerns a ‘construction’ which finally reduces the problem to a topological one. This is then solved by applying a topological fact, Lemma 2.12 below, which follows from Brouwer’s fixed point theorem.

Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 [1] to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.

In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H\in (0,1/2)$. By using the Feynman-Kac representation for the $L^{p}(\Omega )$ moments of the solution, we find the upper and lower bounds for the moments.

We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process $(\xi _{t})$. While the value process $(V_{t})$ of the non-linear problem is only right-uppersemicontinuous, we show that the strict value process $(V^{+}_{t})$ is necessarily right-continuous. Moreover, the strict value process $(V_{t}^{+})$ coincides with the process of right-limits $(V_{t+})$ of the value process. As an auxiliary result, we obtain that a strong non-linear $f$-supermartingale is right-continuous if and only if it is right-continuous along stopping times in conditional $f$-expectation.

In this note, we consider the frog model on $\mathbb {Z}^{d}$ and a two-type version of it with two types of particles. For the one-type model, we show that the asymptotic shape does not depend on the initially activated set and the configuration there. For the two-type model, we show that the possibility for the types to coexist in that both of them activate infinitely many particles does not depend on the choice of the initially activated sets and the configurations there.

We study the asymptotics of Schur polynomials with partitions $\lambda $ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots ,(m-1),0)$ by at most one component at the beginning as $N\rightarrow \infty $, for a positive integer $m\ge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $\frac {1}{N}\log \frac {s_{\lambda }(u_{1},\ldots ,u_{k}, x_{k+1},\ldots ,x_{N})}{s_{\lambda }(x_{1},\ldots ,x_{N})}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_{i}$ for $1\leq i\leq k$, when there are only finitely many distinct $x_{i}$’s and each $u_{i}$ is in a neighborhood of $x_{i}$, as $N\rightarrow \infty $. The results are related to the law of large numbers and central limit theorem for the dimer configurations on contracting square-hexagon lattices with certain boundary conditions.

It is well-known that, in a certain parameter regime, the so-called McKean-Vlasov evolution $(\mu _{t})_{t\in [0,\infty )}$ admits exactly three stationary states. In this paper we study the long-time behaviour of the flow $(\mu _{t})_{t\in [0,\infty )}$ in this regime. The main result is that, for any initial measure $\mu _{0}$, the flow $(\mu _{t})_{t\in [0,\infty )}$ converges to a stationary state as $t\rightarrow \infty $ (see Theorem 1.2). Moreover, we show that if the energy of the initial measure is below some critical threshold, then the limiting stationary state can be identified (see Proposition 1.3). Finally, we also show some topological properties of the basins of attraction of the McKean-Vlasov evolution (see Proposition 1.4). The proofs are based on the representation of $(\mu _{t})_{t\in [0,\infty )}$ as a Wasserstein gradient flow.

Some results of this paper are not entirely new. The main contribution here is to show that the Wasserstein framework provides short and elegant proofs for these results. However, up to the author’s best knowledge, the statement on the topological properties of the basins of attraction (Proposition 1.4) is a new result.

Filter stability is a classical problem in the study of partially observed Markov processes (POMP), also known as hidden Markov models (HMM). For a POMP, an incorrectly initialized non-linear filter is said to be (asymptotically) stable if the filter eventually corrects itself as more measurements are collected. Filter stability results in the literature that provide rates of convergence typically rely on very restrictive mixing conditions on the transition kernel and measurement kernel pair, and do not consider their effects independently. In this paper, we introduce an alternative approach using the Dobrushin coefficients associated with both the transition kernel as well as the measurement channel. Such a joint study, which seems to have been unexplored, leads to a concise analysis that can be applied to more general system models under relaxed conditions: in particular, we show that if $(1 - \delta (T))(2-\delta (Q)) < 1$, where $\delta (T)$ and $\delta (Q)$ are the Dobrushin coefficients for the transition and the measurement kernels, then the filter is exponentially stable. Our findings are also applicable for controlled models.

We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,\eta )$. The main obstacle is the impossibility of averaging $K$-valued random variables. This is overcome by using isometric images of $K$ inside a Banach space or a Hilbert space, after which we can apply results for CLTs on these spaces.

Let $X$ be a simple random walk on $\mathbb {Z}_{n}^{d}$ with $d\geq 3$ and let $t_{\mathrm {cov}}$ be the expected cover time. We consider the set $\mathcal {U}_{\alpha }$ of points of $\mathbb {Z}_{n}^{d}$ that have not been visited by the walk by time $\alpha t_{\mathrm {cov}}$ for $\alpha \in (0,1)$. It was shown in [6] that there exists $\alpha _{1}(d)\in (0,1)$ such that for all $\alpha >\alpha _{1}(d)$ the total variation distance between the law of the set $\mathcal {U}_{\alpha }$ and an i.i.d. sequence of Bernoulli random variables indexed by $\mathbb {Z}_{n}^{d}$ with success probability $n^{-\alpha d}$ tends to $0$ as $n \to \infty $. In [6] the constant $\alpha _{1}(d)$ converges to $1$ as $d\to \infty $. In this short note using the Chen–Stein method and a concentration result for Markov chains of Lezaud [5] we greatly simplify the proof of [6] and find a constant $\alpha _{1}(d)$ which converges to $3/4$ as $d\to \infty $. We prove analogous results for the high points of the Gaussian free field.

Let $(X_{n})_{n\in \mathds {N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb {P}(X_{1}=1)=\mathbb {P}(X_{1}=-1)=1/2$. Let $F(\sigma )=\sum _{n=1}^{\infty }X_{n}n^{-\sigma }$. We prove that the following holds almost surely \[ \limsup _{\sigma \to 1/2^{+}}\frac {F(\sigma )}{\sqrt {2\mathbb {E} F(\sigma )^{2} \log \log \mathbb {E} F(\sigma )^{2}}}=1. \]

We use moment method to understand the cycle structure of the composition of two independent invariant permutations. We prove that under a good control on fixed points and cycles of length $2$, the limiting joint distribution of the number of small cycles is the same as in the uniform case i.e. for any positive integer $k$, the number of cycles of length $k$ converges to the Poisson distribution with parameter $\frac {1}{k}$ and is asymptotically independent of the number of cycles of length $k'\neq k$.

We consider random walks among random conductances on $\mathbb {Z}^{2}$ and establish precise asymptotics for the associated potential kernel and the Green’s function of the walk killed upon exiting balls. The result is proven for random walks on i.i.d. supercritical percolation clusters among ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau $\nabla \phi $-interface model.

We consider the discrete time stopping problem \[ V(t,x) = \sup _{\tau }\mathbb {E}_{(t,x)}g(\tau , X_{\tau }), \] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set $\partial C$. We show that under some conditions $V$ is not smooth in the interior of $C$ either. Even more, under some additional conditions we show that $V$ is not differentiable on a dense subset of $C$. As a guiding example we consider the Chow-Robbins game. We give evidence that $\partial C$ is not smooth and that $C$ is not convex, in the Chow-Robbins game and other examples.

We consider a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in $L$ bins $rL$ balls, where $r$ is a fixed positive constant. At each time step a ball is removed from each non-empty bin. Then all these removed balls are uniformly reassigned into bins. We prove that the mixing time of the RBB process is of order $L$. Furthermore we show that if the initial configuration has $o(L)$ balls per site the equilibrium is attained in $o(L)$ steps.

We use coupling ideas introduced in [13] to show that an IDLA process on a cylinder graph $G\times {\mathbb {Z}} $ forgets a typical initial profile in $\mathcal {O}( N\sqrt {\tau _{N}} (\log \! N)^{2} )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $\tau _{N}$ is the total variation mixing time of a simple random walk on $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb {Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.

Doubly-intractable distributions appear naturally as posterior distributions in Bayesian inference frameworks whenever the likelihood contains a normalizing function $Z$. Having two such functions $Z$ and $\widetilde Z$ we provide estimates of the total variation and $1$-Wasserstein distance of the resulting posterior probability measures. As a consequence this leads to local Lipschitz continuity w.r.t. $Z$. In the more general framework of a random function $\widetilde Z$ we derive bounds on the expected total variation and expected $1$-Wasserstein distance. The applicability of the estimates is illustrated within the setting of two representative Monte Carlo recovery scenarios.

We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and $m$-ary search trees, preferential attachment trees, and others. The limiting constant is computed, analytically or numerically, for several examples. The method is based on Crump–Mode–Jagers branching processes.

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $\partial _{t}u=-(-\Delta )^{\frac {1}{2}}u+\xi u$, where $\xi $ is a periodic spatial white noise. To be precise, we construct limits as $\varepsilon \to 0$ of solutions of $\partial _{t}u_{\varepsilon }=-(-\Delta )^{\frac {1}{2}}u_{\varepsilon }+(\xi _{\varepsilon }-C_{\varepsilon })u_{\varepsilon }$, where $\xi _{\varepsilon }$ is a mollification of $\xi $ at scale $\varepsilon $ and $C_{\varepsilon }$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labbé, “A simple construction of the continuum parabolic Anderson model on $\mathbf {R}^{2}$.”

We prove that the speed of a biased random walk on a supercritical Galton-Watson tree conditioned to survive is analytic within the ballistic regime. This extends the previous work [12] in which it was shown that the speed is differentiable within the range of bias for which a central limit theorem holds.

We consider the classical Skorokhod space ${\mathbb {D}}[0,1]$ and the space of continuous functions ${\mathbb {C}}[0,1]$ equipped with the standard Skorokhod distance $\rho $.

It is well known that neither $({\mathbb {D}}[0,1],\rho )$ nor $({\mathbb {C}}[0,1],\rho )$ is complete. We provide an explicit description of the corresponding completions. The elements of these completions can be regarded as usual functions on $[0,1]$ except for a countable number of instants where their values vary “instantly".

Generators of Markov processes on a countable state space can be represented as finite or infinite matrices. One key property is that the off-diagonal entries corresponding to jump rates of the Markov process are non-negative. Here we present stochastic characterizations of the semigroup generated by a generator with possibly negative rates. This is done by considering a larger state space with one or more particles and antiparticles, with antiparticles being particles carrying a negative sign.

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.

When the distribution of the inter-arrival times of a renewal process is a mixture of geometric laws, we prove that the renewal function of the process is given by the moments of a probability measure which is explicitly related to the mixture distribution. We also present an analogous result in the continuous case when the inter-arrival law is a mixture of exponential laws. We then observe that the above discrete class of renewal processes provides a solvable family of random polymers. Namely, we obtain an exact representation of the partition function of polymers pinned at sites of the aforementioned renewal processes. In the particular case where the mixture measure is a generalized Arcsine law, the computations can be explicitly handled.

We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $\theta $, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $\theta >1$, large deviations events characterized by a small value of $u$, i.e. $u<1-1/\theta $, are such that the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. We generalize these results to the Wishart Ensemble, and we extend them to the first $n$ eigenvalues and the associated eigenvectors.

The Krein condition has been used as a qualitative result to show the M-indeterminacy of some kind of densities. In this work we use results from the theory of the Hilbert transform to construct families of densities having all the same finite moment sequence as a density $f$ with finite logarithmic integral. Actually, our approach explicitly gives Stieltjes classes with center at $f$ and perturbations involving the Hilbert transform of $\ln f$. We consider densities supported on the whole real line or the positive half line.

We prove large deviation results for the position of the rightmost particle, denoted by $M_{n}$, in a one-dimensional branching random walk in a case when Cramér’s condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e. both tails decay as $e^{-\Theta (|t|^{r})}$ for some $r\in ( 0,1)$. It is known that in this case, $M_{n}$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_{n}$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde{M} _{n}$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde{M} _{n}$ as well.

Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller’s coupling, to associate cycles of random permutations governed by the Ewens $(\theta )$ distribution with spacings derived from independent Bernoulli trials with success probability $\theta /(n-1+\theta )$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length $\ell $ are independent Poisson variables with means $\theta /\ell $. Ignatov (1978) first discovered this remarkable result in the uniform case $\theta = 1$, by constructing Bernoulli $(1/n)$ trials as the indicators of record values in a sequence of i.i.d. uniform $[0,1]$ variables. In the present article, the Poisson property of inhomogeneous Bernoulli spacings is explained by a variation of Ignatov’s approach for a general $\theta >0$. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

In this note, we study the model of directed last passage percolation on $\mathbb{Z} ^{2}$, with i.i.d. exponential weight. We consider the maximum directed paths from vertices $(0,\lfloor k^{2/3}\rfloor )$ and $(\lfloor k^{2/3} \rfloor ,0)$ to $(n,n)$, respectively. For the coalescence point of these paths, we show that the probability for it being $>Rk$ far away from the origin is in the order of $R^{-2/3}$. This is motivated by a recent work of Basu, Sarkar, and Sly [7], where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open. Our arguments also apply to other exactly solvable models of last passage percolation.

In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Lévy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.

In this paper we focus on the open symmetric exclusion process with parameter $m$ (open SEP($m/2$)), which allows $m$ particles each site and has an open boundary. We generalize the result about hydrodynamic limit for the open SEP$(m/2)$ originally raised in Theorem 4.12 of [8]. We prove that the hydrodynamic limit of the density profile for a $d$-dimensional open SEP$(m/2)$ solves the $(d+1)$-dimensional heat equation with certain initial condition and boundary condition.

It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.

For a given set of random variables $X_{1},\ldots ,X_{d}$ we seek as large a family as possible of random variables $Y_{1},\ldots ,Y_{d}$ such that the marginal laws and the laws of the sums match: $Y_{i}\,{\stackrel {{d }}{=}}\,X_{i}$ and $\sum _{i}Y_{i}\,{\stackrel {{d }}{=}}\,\sum _{i}X_{i}$. Under the assumption that $X_{1},\ldots ,X_{d}$ are identically distributed but not necessarily independent, using a symmetry-balancing approach we provide a universal construction with sufficient symmetry to satisfy the more stringent requirement that, for any symmetric function $g$, $g(Y)\,{\stackrel {{d }}{=}}\,g(X)$. The same ideas are shown to extend to the non-identically but “similarly” distributed case.

The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: at each time step, two neighbors are randomly chosen and interact if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors’ opinions getting closer to each other. Most of the analytical results established so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space (convex subset and norm) and the initial distribution, but not on the size or the topology of the social network.

In this note, we study a large class of stochastic wave equations with spatial dimension less than or equal to $3$. Via a soft application of Malliavin calculus, we establish that their random field solutions are spatially ergodic.

We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.

We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a ‘generic’ field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.

Given a sequence of Boolean functions $ (f_{n})_{n \geq 1} $, $ f_{n} \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $ (X^{(n)})_{n\geq 1} $ of continuous time $ p_{n} $-biased random walks $ X^{(n)} = (X_{t}^{(n)})_{t \geq 0}$ on $ \{ 0,1 \}^{n} $, let $ C_{n} $ be the (random) number of times in $ (0,1) $ at which the process $ (f_{n}(X_{t}))_{t \geq 0} $ changes its value. In [7], the authors conjectured that if $ (f_{n})_{n \geq 1} $ is non-degenerate, transitive and satisfies $ \lim _{n \to \infty } \mathbb {E}[C_{n}] = \infty $, then $ (C_{n})_{n \geq 1} $ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.

In this article, we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d\geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a comparison principle for multi-type CBI processes being compared to one-dimensional CBI processes, and then an application of an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for the transience of multi-type CBI processes.

We revisit the zero-noise Peano selection problem for Lévy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist. Probab. Lett., 132:62–73, 2018] and show that the selection phenomenon pertains in the multiplicative noise setting and is robust with respect to certain perturbations of the irregular drift and of the small jumps of the noise.

We prove an annealed weak limit of the trajectory of the random walks in cooling random environment (RWCRE) under both slow (polynomial) and fast (exponential) cooling. We identify the weak limit when the underlying static environment is recurrent (Sinai’s model). Avena and den Hollander have previously proved a Gaussian limiting distribution for the distribution of the endpoint of the walk. We find that the weak limit of the trajectory exists as a time-rescaled Brownian motion in the slow cooling case but the limit degenerates to a constant function in the fast cooling one.

We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then we draw a random number of edges from it to existing vertices, chosen independently by the following rule. For each edge, we consider a sample of the growing size of vertices chosen with probabilities proportional to a sublinear function of their degrees. Then the new vertex attaches to the vertex with the highest degree from the sample. Depending on the growth rate of the sample and the sublinear function, the maximum degree could be of sublinear order, of linear order, or having almost all edges drawing to it. The proof uses various stochastic approximation processes and a large deviation approach.

In this note, we prove an $L^{p}$ uniform approximation of the fractional Brownian motion with Hurst exponent $0 < H < \frac {1}{2}$ by means of a family of continuous-time random walks imbedded on a given Brownian motion. The approximation is constructed via a pathwise representation of the fractional Brownian motion in terms of a standard Brownian motion. For an arbitrary choice $\epsilon _{k}$ for the size of the jumps of the family of random walks, the rate of convergence of the approximation scheme is $O(\epsilon _{k}^{p(1-2\lambda )+ 2(\delta -1)})$ whenever $\max \{0,1-\frac {pH}{2}\}< \delta < 1$, $\lambda \in \big (\frac {1-H}{2}, \frac {1}{2} + \frac {\delta -1}{p}\big )$.