Related papers: Existence of generalized Busemann functions and Gi…

We present some new results about the distribution of a random walk whose independent steps follow a $q-$Gaussian distribution with exponent $\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic representation of…

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to…

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment…

We obtain non-Gaussian limit laws for one-dimensional random walk in a random environment assuming that the environment is a function of a stationary Markov process. This is an extension of the work of Kesten, M. Kozlov and Spitzer for…

Linear polymers are represented as chains of hopping reptons and their motion is described as a stochastic process on a lattice. This admittedly crude approximation still catches essential physics of polymer motion, i.e. the universal…

We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder,…

Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $\mu$ such that, for…

We revisit the problem of deriving the mean-field values of avalanche critical exponents in systems with absorbing states. These are well-known to coincide with those of an un-biased branching process. Here, we show that for at least 4…

We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such…

The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization…

We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the…

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations…

Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference…

A particular class of random walks with a spin factor on a three dimensional cubic lattice is studied. This three dimensional random walk model is a simple generalization of random walk for the two dimensional Ising model. All critical…

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff…

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and…

In this paper, we rigorously establish the Gumbel-distributed fluctuations of the cover time, normalized by the mean first passage time, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. This has…

The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset…

In this paper, we obtain a local limit theorem for the Kemperman's model of oscillating random walk on $\mathbb{Z}$; it extends the existing results for classical random walks on $\mathbb Z$ or reflected random walks on $\mathbb N_0$. The…