Related papers: Weakly self-avoiding walk in a pareto-distributed …

In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths,…

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in…

This thesis investigates critical phenomena and equilibrium states in various stochastic models through three interconnected studies. In the first chapter, we analyze the Activated Random Walk model on a one-dimensional ring in the…

Self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters $ \beta_d$ and $ \beta_h $, so that the former controls the distance ($ d $) between the legs…

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a two-state model being a combination of an exploring…

We consider an elementary model for self-organised criticality, the activated random walk on the complete graph. We introduce a discrete time Markov chain as follows. At each time step, we add an active particle at a random vertex and let…

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are…

This paper gives various asymptotic formulae for the transition probability associated with discrete time quantum walks on the real line. The formulae depend heavily on the `normalized' position of the walk. When the position is in the…

Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…

Consider a stochastic process that behaves as a $d$-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the…

We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on $\mathbb Z^d$, $d\ge 2$. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has…

In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The…

The subject of this paper is the simple random walk on $\mathbb{Z}$. We give a very simple answer to the following problem: under the condition that a random walk has already spent $\alpha$-percent of the traveling time on the positive side…

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting…

We study the path behavior of the anisotropic random walk on the two-dimensional lattice Z^2. Simultaneous strong approximations of its components are given.

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…

Let $X$ be a L\'evy process with regularly varying L\'evy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed…

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are…

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length…

We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random mappings in random walks which are shown…