We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.

We develop a Fourier analytic approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate, and it leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Itô and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.

Preferential attachment models with fitness are a substantial extension of the classical preferential attachment model, where vertices have an independent fitness that has a linear impact on its attractiveness in the network formation. As observed by Bianconi and Barabási [4] such network models show different phases. In the condensation phase a small number of exceptionally fit vertices collects a finite fraction of all new links and hence forms a condensate. In this article, we analyse the formation of the condensate for a variant of the model with deterministic normalisation. We consider the regime where the fitness distribution is bounded and has polynomial tail behaviour in its upper end. The central result is a law of large numbers for an appropriately scaled version of the condensate. It follows that a $\Gamma$-distributed shape is formed and, in particular, that the number of vertices contributing to the condensate rises to infinity with increasing network size, in probability.

We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape theorem, and we express the limiting shape in terms of the limiting shapes for the homogeneous processes for the two weight distributions. We further show that there exist pairs of distributions for which the rate of growth in the vertical direction is strictly larger than the rate of growth of the homogeneous process with either of the two distributions, and that this corresponds to the creation of a defect along the vertical axis in the form of a ‘pyramid’.

We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DHN11] and [CS12]. Our result relies on new discrete complex analysis techniques, introduced in [Che12].

We detail some applications, in particular the computation of so-called universal exponents, the proof of quasi-multiplicativity properties of arm probabilities, and bounds on crossing probabilities for the classical Ising model.

We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu [43], as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner [30].

Aging is a prevalent phenomenon in physics, chemistry and many other fields. In this paper we consider the aging process of uncoupled Continuous Time Random Walk Limits (CTRWLs) which are Levy processes time changed by the inverse stable subordinator of index $0 < \alpha < 1$. We apply a recent method developed by Meerscheart and Straka of finding the finite dimensional distributions of CTRWL, to obtaining the aging process’s finite dimensional distributions, self-similarity-like property, asymptotic behavior and its Fractional Fokker-Planck equation(FFPE).

We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.

A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs upto a density for the largest independent set that is bounded by and goes asymptotically to the condensation threshold. We show that the hardcore measure converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tree for which we determine precise asymptotics. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.

We consider measurable $F: \Omega \times \mathbb{R} ^d \to \mathbb{R} $ where for any $x$ the random variable $F(\cdot , x)$ belongs to the Malliavin Sobolev space $\mathbb{D} _{1,2}$ (with respect to a Lévy process) and provide sufficient conditions on $F$ and $G_1,\ldots ,G_d \in \mathbb{D} _{1,2}$ such that $F(\cdot , G_1,\ldots ,G_d) \in \mathbb{D} _{1,2}.$

The above result is applied to show Malliavin differentiability of solutions to BSDEs (backward stochastic differential equations) driven by Lévy noise where the generator is given by a progressively measurable function $f(\omega ,t,y,z).$

We study the asymptotic behaviour of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords and which are coded by stable Lévy processes. We also study other ways to “fill-in” the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.

In this paper, we consider Markov chains of the form $X^{n}_{(k+1)/n}=\psi _{k}(X^{n}_{k/n},Z_{k+1}/\sqrt{n},1/n)$ where the innovation comes from the sequence $Z_{k},k\in \mathbb{N} ^{\ast }$ of independent centered random variables with arbitrary law. Then, we study the convergence $\mathbb{E} [f(X^{n}_t)]\rightarrow \mathbb{E} [f(X_t)]$ where $(X_t)_{t \geqslant 0}$ is a Markov process in continuous time. This may be considered as an invariance principle, which generalizes the classical Central Limit Theorem to Markov chains. Alternatively (and this is the main motivation of our paper), $X^{n}$ may be an approximation scheme used in order to compute $\mathbb{E} [f(X_t)]$ by Monte Carlo methods. Estimates of the error are given for smooth test functions $f$ as well as for measurable and bounded $f.$ In order to prove convergence for measurable test functions we assume that $Z_{k}$ satisfies Doeblin’s condition and we use Malliavin calculus type integration by parts formulas based on the smooth part of the law of $Z_{k}$. As an application, we will give estimates of the error in total variation distance for the Ninomiya Victoir scheme.

We consider the activated random walk model on $\mathbb{Z} ^d$, which undergoes a transition from an absorbing regime to a regime of sustained activity. A central question for this model involves the estimation of the critical density $\mu _c$. We prove that if the jump distribution is biased, then $\mu _c < 1$ for any sleeping rate $\lambda $, $d \geq 1$, and that $\mu _c \to 0$ as $\lambda \to 0$ in one dimension. This answers a question from Rolla and Sidoravicius (2012) and Dickman, Rolla and Sidoravicius (2010) in the case of biased jump distribution. Furthermore, we prove that the critical density depends on the jump distribution.

We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum $\overline Z$, its time $T$, and the process $Z(T+\cdot )-\overline Z$. This expression is in terms of the laws of the original and the tilted Lévy processes conditioned to stay negative and positive respectively. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee’s novel use of Stein’s method for $\{-1,+1\}$ valued variables to a general class of discrete distributions, and provide $\log n$ rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim _{n\to \infty }\frac{X_n} {n}=v_\alpha >0$. Gantert and Zeitouni [9] showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_\omega (X_n < xn)$ with $x \in (0,v_\alpha )$ decay approximately like $\exp \{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-\gamma } \log P_\omega ( X_n < x n)$ converges to $0$ or $-\infty $ depending on whether $\gamma > 1-1/s$ or $\gamma < 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_\omega ( X_n < x n)$ oscillates between $0$ and $-\infty $, almost surely. This had previously been shown only in a very special case of random environments [7].

This paper is concerned with the death-birth updating process. This model is an example of a spatial game in which players located on the $d$-dimensional integer lattice are characterized by one of two possible strategies and update their strategy at rate one by mimicking one of their neighbors chosen at random with a probability proportional to the neighbor’s payoff. To understand the role of space in the form of local interactions, the process is compared with its nonspatial deterministic counterpart for well-mixing populations, which is described by the replicator equation. To begin with, we prove that, provided the range of the interactions is sufficiently large, both strategies coexist on the lattice for a parameter region where the replicator equation also exhibits coexistence. Then, we identify parameter regions in which there is a dominant strategy that always wins on the lattice whereas the replicator equation displays either coexistence or bistability. Finally, we show that, for the one-dimensional nearest neighbor system and in the parameter region corresponding to the prisoner’s dilemma game, cooperators can win on the lattice whereas defectors always win in well-mixing populations, thus showing that space favors cooperation. In particular, several parameter regions where the spatial and nonspatial models disagree are identified.

We prove local existence of solutions for a class of suitably renormalised coupled SPDE–ODE systems driven by space-time white noise, where the space dimension is equal to $2$ or $3$. This class includes in particular the FitzHugh–Nagumo system describing the evolution of action potentials of a large population of neurons, as well as models with multidimensional gating variables. The proof relies on the theory of regularity structures recently developed by M. Hairer, which is extended to include situations with semigroups that are not regularising in space. We also provide explicit expressions for the renormalisation constants, for a large class of cubic nonlinearities.

We propose a simple model of columnar growth through diffusion limited aggregation (DLA). Consider a graph $G_N\times \mathbb{N} $, where the basis has $N$ vertices $G_N:=\{1,\dots ,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. Consider there a simple random walk coming from infinity which deposits on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally.

We prove that there is a critical time scale $N/\log (N)$ for the maximal height of the piles, i.e., there exist constants $\alpha <\beta $ such that the maximal pile height at time $\alpha N/\log (N)$ is of order $\log (N)$, while at time $\beta N/\log (N)$ is larger than $N^\chi $ for some positive $\chi $. This suggests that a monopolistic regime starts at such a time and only the highest pile goes on growing. If we rather consider a walk whose height-component goes down deterministically, the resulting ballistic deposition has maximal height of order $\log (N)$ at time $N$.

These two deposition models, diffusive and ballistic, are also compared with uniform random allocation and Polya’s urn.

Exact upper bounds on the generalized moments $\operatorname{\mathsf {E}} f(S_n)$ of sums $S_n$ of independent nonnegative random variables $X_i$ for certain classes $\mathcal{F} $ of nonincreasing functions $f$ are given in terms of (the sums of) the first two moments of the $X_i$’s. These bounds are of the form $\operatorname{\mathsf {E}} f(\eta )$, where the random variable $\eta $ is either binomial or Poisson depending on whether $n$ is fixed or not. The classes $\mathcal{F} $ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $\operatorname{\mathsf {P}} (S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $\operatorname{\mathsf {E}} \exp \{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.

The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors.

Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a “replica” stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the case of logarithmic average degree.

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators.

The Lamperti–Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, there is a Markov additive process (MAP). In the case that the rssMp is taken to be an $\alpha $-stable process with $\alpha \in (0,2)$, [16] and [24] have computed explicitly the characteristics of the matrix exponent of the semi-group of the embedded MAP, which we henceforth refer to as the Lamperti-stable MAP. Specifically, the matrix exponent of the Lamperti-stable MAP’s transition semi-group can be written in a compact form using only gamma functions.

Just as with Lévy processes, there exists a factorisation of the (matrix) exponents of MAPs, with each of the two factors uniquely characterising the ascending and descending ladder processes, which themselves are again MAPs. Although the case of MAPs with jumps in only one direction should be relatively straightforward to handle, to the author’s knowledge, not a single example of such a factorisation for two-sided jumping MAPs currently exists in the literature. In this article we provide a completely explicit Wiener–Hopf factorisation for the Lamperti-stable MAP.

The main value and novelty of exploring the matrix Wiener–Hopf factorisation of the underlying MAP comes about through style of the computational approach. Understanding the fluctuation theory of the underlying MAP offers new insight into different ways of analysing stable processes. Indeed, we obtain new space-time invariance properties of stable processes, as well as demonstrating examples how new fluctuation identities for stable processes can be developed as a consequence of the reasoning in deriving the matrix Wiener–Hopf factors. The methodology in this paper has already lead to new applications in [27] and [28].

We study the time constant $\mu (e_{1})$ in first passage percolation on $\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $$\lim_{d\to\infty}\frac{\mu({e}_{1})d}{\log d} = \frac{1}{2a},$$ where $a \in [0,\infty ]$ is a constant that depends only on the behavior of the distribution of the passage times at $0$. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a $d$-dimensional cube or diamond, provided that $d$ is large enough.

We study the linear statistics of the circular $\beta $-ensemble with a Stein’s method argument, where the exchangeable pair is generated through circular Dyson Brownian motion. This generalizes previous results obtained in such a way for the CUE and provides a novel approach for studying linear statistics of $\beta $-ensembles. This approach allows studying simultaneously a collection of linear statistics whose number grows with the dimension of the ensemble. Also this approach requires estimating only low order moments of the linear statistics.

We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term $-\Delta \vec \xi $ of the Navier-Stokes equations is substituted by $(-\Delta )^{1+c} \vec \xi $. We investigate how big the correction term $c$ has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as $c>\frac 12$, improving previous results obtained with $c>\frac 32$ in a different setting in [5, 14].

We study the nonlinear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991. This model can be related to the mixing measure of a vertex-reinforced jump process. We prove that the two-point correlation function has a probabilistic interpretation in terms of connectivity in rooted random spanning forests. Using this interpretation, we dominate the two-point correlation function for general pinning, e.g. for uniform pinning, with the corresponding correlation function with pinning at one point. The result holds for a general finite graph, asymptotically as the strength of the pinning converges to zero. Specializing this to general ladder graphs, we deduce in the same asymptotic regime exponential decay of correlations for general pinning.

We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime and then splits into two new particles and, when a particle is at a distance $L$ of the highest particle, it dies without splitting. This model has been introduced by Brunet, Derrida, Mueller and Munier [10] in the physics literature and is called the $L$-branching Brownian motion. We show that the position of the system grows linearly at a velocity $v_L$ almost surely and we compute the asymptotic behavior of $v_L$ as $L$ tends to infinity: \[v_L = \sqrt{2} - \frac{\pi ^2} {2\sqrt{2} L^2} + o\left({\frac {1}{L^2}}\right) ,\] as conjectured in [10]. The proof makes use of results by Berestycki, Berestycki and Schweinsberg [5] concerning branching Brownian motion in a strip.

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda ^{|\sigma |}$ where $\lambda >0$ is a parameter and $|\sigma |$ denotes the total edge length of the triangulation. When $\lambda \in (0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp (\Omega (n^2))$ for $\lambda >1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $\lambda =1$.

We present results concerning when the joint distribution of an exchangeable sequence is determined by the marginal distributions of its partial sums. The question of whether or not this determination occurs was posed by David Aldous. We then consider related uniqueness problems, including a continuous time analog to the Aldous problem and a randomized univariate moment problem.

We consider the contact process with infection rate $\lambda $ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $\tau _{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $\lambda $ is smaller or larger than $\lambda _1(\mathbb{T} ^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $\lambda < \lambda _1(\mathbb{T} ^d)$, $\tau _{G_n}$ grows logarithmically with $n$, while if $\lambda > \lambda _1(\mathbb{T} ^d)$, it grows exponentially with $n$. This result differs from the situation where, instead of $G_n$, the contact process is considered on the $d$-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on $\mathbb{T} ^d$.

We prove a large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails, that is, the distribution tails of the diagonal entries $\mathbb{P} ( |X_{1,1}|>t)$ and off-diagonal entries $\mathbb{P} (|X_{1,2}|>t)$ behave like $e^{-bt^{\alpha }}$ and $e^{-at^{\alpha }}$ respectively, for some $a,b\in (0,+\infty )$ and $\alpha \in (0,2)$. The large deviations principle is of speed $N^{\alpha /2}$, and with a good rate function depending only on the distribution tail of the entries.

We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an integrability assumption. For the proof we use Davies’ perturbation method, where we show a maximal inequality for the perturbed heat kernel via Moser iteration.

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter $H ~(1/3 < H \le 1/2)$ under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe’s distributional Malliavin calculus.

In the present article we consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in [9]. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*} $, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system, nor that the filtration is quasi left-continuous. In particular, it generalizes the Doob-Meyer decomposition of Mertens [36] for classical supermartingales, as well as Peng’s [41] version for right-continuous $g$-supermartingales. As examples of application, we prove an optional decomposition theorem for $g$-supermartingale systems, and also obtain a general version of the well-known dual formulation for BSDEs with constraint on the gains-process, using very simple arguments.

We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian forcing. However, the analysis based on the study of one-sided minimizers of the relevant action is different. In contrast with previous work, finite time coalescence of the minimizers does not hold, and hyperbolicity (exponential convergence of minimizers in reverse time) is not known. In order to establish a One Force — One Solution principle on each ergodic component, we use an extremely soft method to prove a weakened hyperbolicity property and to construct Busemann functions along appropriate subsequences.

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.

Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

We prove a local limit theorem for sums of independent random vectors satisfying appropriate tightness assumptions. In particular, the local limit theorem holds in dimension 1 if the summands are uniformly bounded.

Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions. In particular this covers random permutations generated from Mallows Model with Kendall’s Tau, $\mu $ random permutations introduced in [11], as well as a class of exponential families introduced in [15].

We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlevé-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.

Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda $ if it jumps at rate $\lambda $ according to the matrix $P$. Fix $\lambda _X,\lambda _Y,\lambda _Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $\lambda _X,\lambda _Y$ and $\lambda _Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $\lambda _X,\lambda _Y$ and $\lambda _Z$ with $\max (\lambda _X,\lambda _Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $\lambda _X=\lambda _Y=1$ and $\lambda _Z=0$ we prove a strengthening of our main theorem using a martingale argument. We provide an example showing the transitivity assumption cannot be removed for general $\lambda _X,\lambda _Y$ and $\lambda _Z$.

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of a certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to a decomposition of a time-changed Brownian motion $\mathbb{B} (t^\alpha ), \alpha \in (0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of a fractional Brownian motion with Hurst index $H\in (0,1/2)$. The decomposition in the latter case is a special case of the decomposition of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as a correlated random walk, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.

The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple random walk on a sequence of uniformly bounded degree expander graphs. These examples give a negative answer to a question of Ding, Lubetzky and Peres

In [3], exclusion sensitivity and exclusion stability for symmetric exclusion processes on graphs were defined as natural analogues of noise sensitivity and noise stability in this setting. As these concepts were defined for any sequence of connected graphs, it is natural to study the monotonicity properties of these definitions, and in particular, if some graphs are in some sense more stable or sensitive than others. The main purpose of this paper is to answer one such question which was stated explicitly in [3]. In addition, we get results about the eigenvectors and eigenvalues of symmetric exclusion processes on complete graphs.

Let $\mathcal{K} \subset R^d$, $d\ge 2$, be a smooth, bounded domain satisfying $0\in \mathcal{K} $, and let $f(t),\ t\ge 0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)\mathcal{K} \subset R^d$. Consider a diffusion process corresponding to the generator $\frac 12\Delta +b(x)\nabla $ in the time-dependent region $\bar D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac 12\frac{d^2} {dx^2}+B(x)\frac d{dx}$ on the time-dependent region $[1,f(t)]$ with reflection at the boundary. We give precise conditions for transience/recurrence of the one-dimensional process in terms of the growth rates of $B(x)$ and $f(t)$. In the recurrent case, we also investigate positive recurrence, and in the transient case, we also consider the asymptotic growth rate of the process. Using the one-dimensional results, we give conditions for transience/recurrence of the multi-dimensional process in terms of the growth rates of $B^+(r)$, $B^-(r)$ and $f(t)$, where $B^+(r)=\max _{|x|=r}b(x)\cdot \frac x{|x|}$ and $B^-(r)=\min _{|x|=r}b(x)\cdot \frac x{|x|}$.

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large $\beta $). The main results are bounds on percolation thresholds $\rho _\pm (\beta )$ in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures

This paper addresses the question of predicting when a positive self-similar Markov process $X$ attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in [9] under the assumption that $X$ is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to [3], where the same question is studied for a Lévy process drifting to $-\infty $. The connection to [3] relies on the so-called Lamperti transformation [15] which links the class of positive self-similar Markov processes with that of Lévy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity.

This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk. The intensity of the interaction between monomers is denoted by $\beta \in (0,\infty )$ and there exists a critical threshold $\beta _c$ which determines the three regimes displayed by the model, i.e., extended for $\beta <\beta _c$, critical for $\beta =\beta _c$ and collapsed for $\beta >\beta _c$.

In [4], physicists displayed some numerical results concerning the typical growth rate of some geometric features of the path as its length $L$ diverges. From this perspective the quantities of interest are the horizontal extension of the path and its lower and upper envelopes.

With the help of a new random walk representation, we proved in [10] that the path grows horizontally like $\sqrt{L} $ in its collapsed regime and that, once rescaled by $\sqrt{L} $ vertically and horizontally, its upper and lower envelopes converge towards some deterministic Wulff shapes.

In the present paper, we bring the geometric investigation of the path several steps further. In the collapsed regime, we identify the joint limiting distribution of the fluctuations of the upper and lower envelopes around their associated limiting Wulff shapes, rescaled in time by $\sqrt{L} $ and in space by $L^{1/4}$. In the critical regime we identify the limiting distribution of the horizontal extension rescaled by $L^{2/3}$ and we show that the excess partition function decays as $L^{2/3}$ with an explicit prefactor. In the extended regime, we prove a law of large number for the horizontal extension of the polymer rescaled by its total length $L$, we provide a precise asymptotics of the partition function and we show that its lower and upper envelopes, once rescaled in time by $L$ and in space by $\sqrt{L} $, converge towards the same Brownian motion.

In this paper, we study the overlap distribution and Gibbs measure of the Branching Random Walk with Gaussian increments on a binary tree. We first prove that the Branching Random Walk is 1 step Replica Symmetry Breaking and give a precise form for its overlap distribution, verifying a prediction of Derrida and Spohn. We then prove that the Gibbs measure of this system satisfies the Ghirlanda-Guerra identities. As a consequence, the limiting Gibbs measure has Poisson-Dirichlet statistics. The main technical result is a proof that the overlap distribution for the Branching Random Walk is supported on the set $\{0,1\}$.

We give an algorithm to construct a translation-invariant transport kernel between two arbitrary ergodic stationary random measures on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an arbitrary ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given a pair of realizations of the two measures. The (non-constructive) existence of such a transport kernel was proved in [9]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained transport densities and transport kernels. We give a definition of stability of constrained transport densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained transport densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.

For beta-ensembles with convex polynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.

We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate $b$. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\theta $ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new type, called allele, to his carrier. We study the allele frequency spectrum which is the numbers $A(k,t)$ of types represented by $k$ alive individuals in the population at time $t$. Thanks to a new construction of the coalescent point process describing the genealogy of individuals in the splitting tree, we are able to compute recursively all joint factorial moments of $\left (A(k,t)\right )_{k\geq 1}$. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.

We consider the uniform infinite quadrangulation of the plane (UIPQ). Curien, Ménard and Miermont recently established that in the UIPQ, all infinite geodesic rays originating from the root are essentially similar, in the sense that they have an infinite number of common vertices. In this work, we identify the limit quadrangulation obtained by rerooting the UIPQ at a point at infinity on one of these geodesics. More precisely, calling $v_k$ the $k$-th vertex on the “leftmost” geodesic ray originating from the root, and $Q_{\infty }^{(k)}$ the UIPQ re-rooted at $v_k$, we study the local limit of $Q_{\infty }^{(k)}$. To do this, we split the UIPQ along the geodesic ray $(v_k)_{k\geq 0}$. Using natural extensions of the Schaeffer correspondence with discrete trees, we study the quadrangulations obtained on each “side” of this geodesic ray. We finally show that the local limit of $Q_{\infty }^{(k)}$ is the quadrangulation obtained by gluing the limit quadrangulations back together.

We prove that any rooted stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to random infinite planar triangulations and Delaunay graphs are given.

We use the connection between bond percolation and SIR epidemics to establish lower bounds for the critical percolation probability in $2$ and $3$ dimensions as the range becomes large. The bound agrees with the conjectured asymptotics for the long range critical probability, refines results of M. Penrose, and complements results of van der Hofstad and Sakai in dimensions greater than $6$.

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point but their formula is not completely correct.

We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\mathbb{Z}$. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on $\mathbb{R}$, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as functionals of the Brownian web, and furthermore, we show that its evolution solves a martingale problem. The generator for the continuum-sites stepping stone model has a singular feature: at each time, the resampling of genealogies only affects a set of individuals of measure $0$. Along the way, we prove some negative correlation inequalities for coalescing Brownian motions, as well as extend the theory of marked metric measure spaces (developed recently by Depperschmidt, Greven and Pfaffelhuber [DGP11]) from the case of probability measures to measures that are finite on bounded sets.

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $\gamma $-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.

We introduce the “mass migration process” (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\geq 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We derive necessary and sufficient conditions for the existence of translation invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation invariant, invariant probability measures. We also consider attractiveness properties (through couplings), condensation phenomena, and their links for MMP. We illustrate our results on many examples; we prove the coexistence of condensation and attractiveness in one of them.

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log (\alpha )/\alpha $ for a large selection coefficient $\alpha $. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $\mu $ for which the fixation times have different asymptotics as $\alpha \to \infty $.

If $\mu $ is of order $\alpha $, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log (\alpha )/\alpha $. If $\mu $ is of order $\alpha ^\gamma , 0\leq \gamma \leq 1$, the fixation time is $\sim (2 + (1-\gamma )\Delta ) \log (\alpha )/\alpha $, where $\Delta $ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $\mu = 1/\log (\alpha )$, the fixation time is $\sim (2+S)\log (\alpha )/\alpha $, where $S$ is a random time in a simple epidemic model.

The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone’s ancestral selection graph.

We study scaling limits of a Laplacian pinning model in $(1+1)$ dimension and derive sample path large deviations for the profile height function. The model is given by a Gaussian integrated random walk (or a Gaussian integrated random walk bridge) perturbed by an attractive force towards the zero-level. We study in detail the behaviour of the rate function and show that it can admit up to five minimisers depending on the choices of pinning strength and boundary conditions. This study complements corresponding large deviation results for Gaussian gradient systems with pinning in $ (1+1) $-dimension ([FS04]) in $(1+d) $-dimension ([BFO09]), and recently in higher dimensions in [BCF14].

Consider a random trigonometric polynomial $X_n: \mathbb{R} \to \mathbb{R} $ of the form \[ X_n(t) = \sum _{k=1}^n \left ( \xi _k \sin (kt) + \eta _k \cos (kt)\right ), \] where $(\xi _1,\eta _1),(\xi _2,\eta _2),\ldots $ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in \mathbb{N} }$ be any sequence of real numbers. We prove that as $n\to \infty $, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correlation function $(\sin t)/t$. We also establish similar local universality results for centered random vectors $(\xi _k,\eta _k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $\alpha $-stable law.

We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi , \zeta $ be two RCLL adapted processes with $\xi \leq \zeta $. The criterium is given by \[ {\cal J}_{\tau , \sigma }= {\cal E}^g_{0, \tau \wedge \sigma } \left (\xi _{\tau }\textbf{1} _{\{ \tau \leq \sigma \}}+\zeta _{\sigma }\textbf{1} _{\{\sigma <\tau \}}\right ), \] where $\tau $ and $ \sigma $ are stopping times valued in $[0,T]$. Under Mokobodzki’s condition, we establish the existence of a value function for this game, i.e. $\inf _{\sigma }\sup _{\tau } {\cal J}_{\tau , \sigma } = \sup _{\tau } \inf _{\sigma } {\cal J}_{\tau , \sigma }$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi $ and $\zeta $ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

The Euclidean first-passage percolation (FPP) model of Howard and Newman is a rotationally invariant model of FPP which is built on a graph whose vertices are the points of homogeneous Poisson point process. It was shown by Howard-Newman that one has (stretched) exponential concentration of the passage time $T_n$ from $0$ to $n\mathbf{e} _1$ about its mean on scale $\sqrt{n} $, and this was used to show the bound $\mu n \leq \mathbb{E} T_n \leq \mu n + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy between the expected passage time and its deterministic approximation $\mu = \lim _n \frac{\mathbb {E}T_n} {n}$. In this paper, we introduce an inductive entropy reduction technique that gives the stronger upper bound $\mathbb{E} T_n \leq \mu n + C_k\psi (n) \log ^{(k)}n$, where $\psi (n)$ is a general scale of concentration and $\log ^{(k)}$ is the $k$-th iterate of $\log $. This gives evidence that the inequality $\mathbb{E} T_n - \mu n \leq C\sqrt{\mathrm {Var}~T_n} $ may hold.

We study i.i.d. sums $\tau _k$ of nonnegative variables with index $0$: this means ${\mathbf P}(\tau _1=n) = {\varphi }(n) n^{-1}$, with ${\varphi }(\cdot )$ slowly varying, so that ${\mathbf E}(\tau _1^{\epsilon })=\infty $ for all ${\epsilon }>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of ${\mathbf P}(\tau _k=n)$ when $n$ is at least the typical length of $\tau _k$. A recent renewal theorem in [22] is an immediate consequence: ${\mathbf P}(n\in \tau ) \sim{\mathbf P} (\tau _1=n)/{\mathbf P}(\tau _1 > n)^2$ as $n\to \infty $. If instead we only assume regular variation of ${\mathbf P}(n\in \tau )$ and slow variation of $U_n:= \sum _{k=0}^n {\mathbf P}(k\in \tau )$, we obtain a similar equivalence but with ${\mathbf P}(\tau _1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of ${\mathbf P}(\tau _k \leq n)$ when $n$ is much smaller than the typical length of $\tau _k$.

A $d$-dimensional branching diffusion, $Z$, is investigated, where the linear attraction or repulsion between particles is competing with an Ornstein-Uhlenbeck drift, with parameter $b$ (we take $b>0$ for inward O-U and $b<0$ for outward O-U). This work has been motivated by [4], where a similar model was studied, but without the drift component.

We show that the large time behavior of the system depends on the interaction and the drift in a nontrivial way. Our method provides, inter alia, the SLLN for the non-interactive branching (inward) O-U process.

First, regardless of attraction ($\gamma >0$) or repulsion ($\gamma <0$), a.s., as $t\to \infty $, the center of mass of $Z_t$ converges to the origin when $b>0$, while escapes to infinity exponentially fast (rate $|b|$) when $b<0$.

We then analyze $Z$ as viewed from the center of mass, and finally, for the system as a whole, we provide a number of results and conjectures regarding the long term behavior of the system; some of these are scaling limits, while some others concern local extinction.

We study the intersection of two independent renewal processes, $\rho =\tau \cap \sigma $. Assuming that ${\mathbf P}(\tau _1 = n ) = \varphi (n)\, n^{-(1+\alpha )}$ and ${\mathbf P}(\sigma _1 = n ) = \tilde \varphi (n)\, n^{-(1+ \tilde \alpha )} $ for some $\alpha ,\tilde{\alpha } \geq 0$ and some slowly varying $\varphi ,\tilde \varphi $, we give the asymptotic behavior first of ${\mathbf P}(\rho _1>n)$ (which is straightforward except in the case of $\min (\alpha ,\tilde \alpha )=1$) and then of ${\mathbf P}(\rho _1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities ${\mathbf P}(\rho _1=n)$ while knowing asymptotically the renewal mass function ${\mathbf P}(n\in \rho )={\mathbf P}(n\in \tau ){\mathbf P}(n\in \sigma )$. Our results can be used to bound coupling-related quantities, specifically the increments $|{\mathbf P}(n\in \tau )-{\mathbf P}(n-1\in \tau )|$ of the renewal mass function.

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.

We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results in [DK12] for excited random walks with “boundedly many cookies per site.” In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many “cookies” at each site. Moreover, a BMPE has two parameters $\alpha ,\beta <1$ and the scaling limits in this paper cover a larger variety of choices for $\alpha $ and $\beta $ than can be obtained for ERWs with boundedly many cookies per site.

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the “environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a “locally strong” perturbation of a simple random walk by a random potential induced by “rare impurities,” which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai’s scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha ,$ where $\alpha >0$ is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [29] and [31].

This article revisits the work by Ofer Zeitouni and Steve Zelditch on large deviations for the empirical measures of random orthogonal polynomials with i.i.d. Gaussian complex coefficients, and extends this result to real Gaussian coefficients. This article does not require any knowledge in geometry. For clarity, we focus on two classical cases: Kac polynomials and elliptic polynomials.