We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections, drift), we study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies between the angles of the model.

A complicated integral expression for this Laplace transform has recently been obtained by two authors of this paper. In the differentially algebraic case, we provide a simple, explicit integral-free expression in terms of a hypergeometric function. It specializes to earlier expressions in several classical cases: the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case (corresponding to the so-called Dieker-Moriarty conditions on the parameters). This paper thus closes, in a sense, the quest of all “simple” cases.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte’s invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform (or its square) as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

We consider homogenization of random surfaces and study the variational principle for graph homomorphisms from subsets of ${\mathbb{Z}}^m$ into $\mathbb{Z}$, where the underlying uniform measure is perturbed by a random potential. Motivated by the theories of random walks in random potentials, we assume that random potential is stationary, ergodic, and bounded in $L^1$. We show that the variational principle holds in probability and that the entropy functional homogenizes, i.e. is independent of the values taken by the random potential. The main ingredients in the argument are the existence of the quenched surface tension, the equivalence of the quenched and the annealed surface tension, and robustness of the surface tension under change in boundary data. These ingredients are deduced by a combination of a superadditive ergodic theorem and combinatorial results, especially the Kirszbraun theorem.

We study Brownian paths perturbed by semibounded pair potentials and prove upper bounds on the mean square displacement. As a technical tool we derive infinite dimensional versions of key inequalities that were first used in [15] in order to study the effective mass of the Fröhlich polaron.

This paper investigates the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant ${\mathbb{Z}}_{+}^2$. There are four possible distributions for the jumps of these processes, depending on the location of the starting point: in the interior, on the two positive axes of the boundary, and at the origin $(0,0)$.

With mild conditions on the positive jumps of the random walk, which can be unbounded, a complete analysis of the asymptotic behavior of the Green function of the random walk killed at $(0,0)$ is achieved. The main result is that eight regions of the set of parameters determine completely the possible limiting behaviors of Green functions of these Markov chains. These regions are defined by a set of relations for several characteristics of the distributions of the jumps.

In the transient case, a description of the Martin boundary is obtained and in the positive recurrent case, our results give the exact limiting behavior of the invariant distribution of a state whose norm goes to infinity along some asymptotic direction in the quadrant. These limit theorems extend results of the literature obtained, up to now, essentially for random walks whose jump sizes are either 0 or 1 on each coordinate.

Our approach relies on a combination of several methods: probabilistic representations of solutions of analytical equations, Lyapounov functions, convex analysis, methods of homogeneous random walks, and complex analysis arguments.

We consider real, Gauss-divisible matrices $A_t=A+\sqrt{t}B$, where B is from the real Ginibre ensemble. We prove that the bulk correlation functions converge to a universal limit for $t=O(N^{-1∕3+\mathit{ϵ}})$ if A satisfies certain local laws. If $A=\frac{1}{\sqrt{N}}{({\mathrm{\xi }}_{jk})}_{j,k=1}^N$ with ${\mathrm{\xi }}_{jk}$ independent and identically distributed real random variables having zero mean, unit variance and finite moments, the Gaussian component can be removed using local laws proven by Bourgade–Yau–Yin, Alt–Erdős–Krüger and Cipolloni–Erdős–Schröder and the four moment theorem of Tao–Vu.

We explore the voter model dynamics on a directed random graph model ensemble (digraphs), given by the Directed Configuration Model. The voter model captures the evolution of opinions over time on a graph where each vertex represents an individual holding a binary opinion. Our primary interest lies in the density of discordant edges, defined as the fraction of edges connecting vertices with different opinions, and its asymptotic behaviour as the graph size grows to infinity. This analysis provides valuable insights, not only into the consensus time behaviour but also into how the process approaches this absorption time on shorter time scales. Our analysis is based on the study of certain annealed random walk processes evolving on out-directed, marked Galton-Watson trees, which describe the locally tree-like nature of the considered random graph model. Additionally, we employ innovative coupling techniques that exploit the classical stochastic dual process of coalescing random walks. We extend existing results on random regular graphs to the more general setting of heterogeneous and directed configurations, highlighting the role of graph topology in the opinion dynamics.

In this paper we study the recurrence and transience of the ${\mathbb{Z}}^d$-valued branching random walk in random environment indexed by a critical Bienaymé-Galton-Watson tree, conditioned to survive. The environment is made either of random conductances or of random traps on each vertex. We show that when the offspring distribution is non degenerate with a finite third moment and the environment satisfies some suitable technical assumptions, then the process is recurrent up to dimension four, and transient otherwise. The proof is based on a truncated second moment method, which only requires to have good estimates on the quenched Green’s function.