Related papers: The odometer in subcritical activated random walk

We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb{Z}^d$, $d\geq 3$, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost…

We consider a branching random walk on $\mathbb{R}$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that the tail distribution of $Z$ displays a…

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed…

We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for the position of the walker, properly…

We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on…

We consider a system of asymmetric independent random walks on $\mathbb{Z}^d$, denoted by $\{\eta_t,t\in{\mathbb{R}}\}$, stationary under the product Poisson measure $\nu_{\rho}$ of marginal density $\rho>0$. We fix a pattern $\mathcal{A}$,…

Continuous-time random walks offer powerful coarse-grained descriptions of transport processes. We here microscopically derive such a model for a Brownian particle diffusing in a deep periodic potential. We determine both the waiting-time…

The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented…

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove…

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…

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$…

We introduce the concept of a deterministic walk. Confining our attention to the finite state case, we establish hypotheses that ensure that the deterministic walk is transitive, and show that this property is in some sense robust. We also…

Motivated by the dissipative abelian sandpile model, we analyze the trajectories of a one-dimensional random walk in a landscape of soft traps. These traps, placed at increasing distances from each other, correspond to dissipative sites in…

We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin $O(N)$ model on the torus of $\mathbb{Z}^d$, $d \geq 3$, when $N \in \mathbb{N}_{>0}$ and the inverse temperature $\beta$ is large…

We obtain expected number of arrivals, absorption probabilities and expected time until absorption for an asymmetric discrete random walk on a graph in the presence of multiple function barriers. On each edge of the graph and in each vertex…

We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that if the random walk is…

We introduce a family of atomic measures on free groups generated by no-return random walks. These measures are shown to be very convenient for comparing "relative sizes" of subgroups, context-free and regular subsets (that, subsets…

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d larger than 1), are named Dirichlet when their step lengths are distributed according to a Dirichlet law. The latter continuous…

We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an $\ell^\infty$-infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a…

We report on a closed-form expression for the survival probability of a discrete 1D biased random walk to not return to its origin after N steps. Our expression is exact for any N, including the elusive intermediate range, thereby allowing…