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Let $S$ be a set of $n$ points in $d$-dimensional Euclidean space. Assign to each $x\in S$ an arbitrary distance $r(x)>0$. Let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\"os and Pach (1988)…

Consider the time T_oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T_oz in terms of the volume of z and the graph distance between o and z. The bounds are for…

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\Z^d$, RWDE are parameterized…

We are studying the motion of a random walker in two and three dimensional continuum with uniformly distributed jump-length. This is different from conventional Lavy flight. In 2D and 3D continuum, a random walker can move in any direction,…

One dimensional excited random walk has been extensively studied for bounded, i.i.d. cookie environments. In this case, many important properties of the walk including transience or recurrence, positivity or non-positivity of the speed, and…

We introduce a formalism based on a continuous time approximation, to study the characteristics of Page Rank random walks. We find that the diffusion of the occupancy probability has a dynamics that exponentially "forgets" the initial…

Sites in an infinite d-dimensional lattice, open with probability greater or equal to 1/d, form an infinite open path.

We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length $n$, comes…

A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…

For a graph $G$ on $n$ vertices, naively sampling the position of a random walk of at time $t$ requires work $\Omega(t)$. We desire local access algorithms supporting $\text{position}(G,s,t)$ queries, which return the position of a random…

In a Markov chain started at a state $x$, the hitting time $\tau(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(\tau(y) = t)$ that the first visit to $y$ occurs precisely at a given…

In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate results were needed concerning the recurrence properties of its unperturbed version. These were analogous to the…

Excited random walk is a random walk that has a positive drift to the right when it reaches a vertex it hasn't been to before. We show that in three dimensions the walk drifts to the right in non-zero speed.

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a…

We investigate the first passage statistics of active continuous time random walks with Poissonian waiting time distribution on a one dimensional infinite lattice and a two dimensional infinite square lattice. We study the small and large…

For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{Z}^d$. Define the $\alpha$-fold self intersection local time $L_n(\alpha) := \sum_{x} l(n,x)^{\alpha}$, and let…

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…

Majumdar and Tamm [Phys. Rev. E 86 021135 (2012), arXiv:1206.6184] recently obtained analytical expressions for the mean number of common sites W_N(t) visited up to time t by N independent random walkers starting from the origin of a…

For any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on…

We consider a class of self-interacting random walks in deterministic or random environments, known as excited random walks or cookie walks, on the d-dimensional integer lattice. The main purpose of this paper is two-fold: to give a survey…