Related papers: Favorite sites for simple random walk in two and m…

For a one-dimensional simple random walk $(S_t)$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work…

In this paper, we study favorite sites of one-dimensional asymmetric simple random walks. We show that almost surely, for any fixed integer $r\geq 1$, ``$r$ favorite sites" occurs infinitely often. We also give the asymptotic growth rate of…

For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with…

We prove that, with probability one, eventually there are no more than three favourite (i.e. most visited) sites of simple random walk. This partially answers a relatively long standing question of Pal Erdos and Pal Revesz.

In this article, we study special points of a simple random walk and a Gaussian free field, such as (nearly) favorite points, late points and high points. In section $2$, we extend results of [19] and suggest open problems for $d=2$. In…

Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local…

We investigate a problem suggested by Dembo, Peres, Rosen, and Zeitouni, which states that the growth exponent of favorite points associated with a simple random walk in ${\mathbb Z}^2$ coincides, on average and almost surely, with those of…

Erd\H{o}s and R\'ev\'esz initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical…

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…

For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the…

Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show…

We study the favourite sites of a random walk evolving in a sparse random environment on the set of integers. The walker moves symmetrically apart from some randomly chosen sites where we impose random drift. We prove annealed limit…

We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite…

Random walks with memory typically involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the statistics of first-passage and…

We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pal Erdos and Pal Revesz in the early eighties.

For a simple symmetric random walk in dimension $d\geq 3$, a uniform strong law of large numbers is proved for the number of sites with given local time up to time $n$.

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random…

Consider an arbitrary transient random walk on $\Z^d$ with $d\in\N$. Pick $\alpha\in[0,\infty)$ and let $L_n(\alpha)$ be the spatial sum of the $\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range,…

In the classical paper of Dvoretzky-Erd\H{o}s, asymptotics for the expected value and the variance of the number of distinct sites visited by a Simple Symmetric Random Walk were calculated. Here, these results are generalized for Random…

Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at…