Related papers: Exceptional Times and Invariance for Dynamical Ran…

We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincar\'e inequality w.r.t. $\mu$. The random walk…

We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. Homogenization and regeneration techniques combine to prove a law of large numbers and an averaged invariance…

Two fundamental theorems by Spitzer/Erickson and Kesten/Maller on the fluctuation type (positive divergence, negative divergence or oscillation) of a real-valued random walk $(S_{n})_{n\ge 0}$ with iid increments $X_{1},X_{2},\ldots$ and…

We consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization,…

We study random walks on the integers driven by a sample of time-dependent nearest-neighbor conductances that are bounded but are permitted to vanish over time intervals of positive Lebesgue-length. Assuming only ergodicity of the…

We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…

Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is…

Let $W$ be an integer valued random variable satisfying $E[W] =: \delta \geq 0$ and $P(W<0)>0$, and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any…

The dynamical discrete web (DDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical parameter s. The evolution is by…

Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly)…

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of…

The longest increasing subsequence (LIS) of a random walk has so far been studied mainly for zero-mean, symmetric step increments. We numerically investigate the LIS of biased Gaussian random walks, with unit-variance increments and…

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk…

We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over…

In this paper we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an iid fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary…

We study a model of multi-excited random walk with non-nearest neighbour steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either $x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the multi-type…

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random…

Let $S(n)$ be a centered random walk with finite second moment. We consider the integrated random walk $T(n) = S(0)+S(1)+\dots+S(n)$. We prove invariance principles for the meander and for the bridge of this process, under the condition…

Discrete time random walks, in which a step of random sign but constant length $\delta x$ is performed after each time interval $\delta t$, are widely used models for stochastic processes. In the case of a correlated random walk, the next…