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In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and…

Probability · Mathematics 2017-07-05 Gideon Amir , Itai Benjamini , Ori Gurel-Gurevich , Gady Kozma

A rotor configuration on a graph contains in every vertex an infinite ordered sequence of rotors, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a rotor walk is a…

Probability · Mathematics 2017-07-05 Sebastian Mueller , Tal Orenshtein

In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $\alpha\in (1, 2)$ and obtain the asymptotic form of the…

Probability · Mathematics 2018-08-07 Kohei Uchiyama

To what extent is the underlying distribution of a finitely supported unbiased random walk on $\mathbb{Z}$ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various…

Probability · Mathematics 2025-12-02 Michael J. Larsen

We study random walks in a random environment on a regular, rooted, coloured tree. The asymptotic behaviour of the walks is classified for ergodicity/transience in terms of the geometric properties of the matrix describing the random…

Probability · Mathematics 2007-05-23 Mikhail Menshikov , Dimitri Petritis

We study a discrete-time random walk on the non-negative integers, such that when 0 is reached a jump occurs to an arbitrary location, with given probabilities. We obtain an asymptotic formula for the expected position at large times, in…

Probability · Mathematics 2011-09-01 Guy Katriel

We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is…

Probability · Mathematics 2017-06-19 Amichai Lampert , Assaf Shapira

We introduce a set of techniques that allow for efficiently generating many independent random walks in the Massive Parallel Computation (MPC) model with space per machine strongly sublinear in the number of vertices. In this…

Data Structures and Algorithms · Computer Science 2019-11-07 Jakub Łącki , Slobodan Mitrović , Krzysztof Onak , Piotr Sankowski

We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful…

Combinatorics · Mathematics 2007-05-23 Joshua N. Cooper , Joel Spencer

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…

Statistical Mechanics · Physics 2022-11-23 E. Ben-Naim , P. L. Krapivsky

Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$…

Probability · Mathematics 2024-11-22 Stein Andreas Bethuelsen , Florian Völlering

Applied to statistical physics models, the random cost algorithm enforces a Random Walk (RW) in energy (or possibly other thermodynamic quantities). The dynamics of this procedure is distinct from fixed weight updates. The probability for a…

Statistical Mechanics · Physics 2009-10-31 Bernd A. Berg , Ulrich H. E. Hansmann

Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic…

Probability · Mathematics 2015-12-14 Lorenz A. Gilch , François Ledrappier

We consider a random walk in random environment in the low disorder regime on $\mathbb Z^d$. That is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\epsilon \xi(x,e)$, where…

Probability · Mathematics 2015-11-11 David Campos , Alejandro F. Ramirez

Given a finite-range random walk on a finitely generated free group , what is the asymptotic behaviour, as the number of steps goes to infinity, of the sequence of probabilities that the random walk is at a given element of the group? In…

Probability · Mathematics 2025-07-22 Guillaume Chevalier

In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We…

Probability · Mathematics 2020-11-23 Tabea Glatzel , Jan Nagel

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a "mating-of-trees" type bijection. This class includes the…

Probability · Mathematics 2020-08-27 Ewain Gwynne , Jason Miller

We study an intermittent random walk on a random network of scale-free degree distribution. The walk is a combination of simple random walks of duration $t_w$ and random long-range jumps. While the time the walker needs to cover all the…

Disordered Systems and Neural Networks · Physics 2015-06-25 A. Ramezanpour

We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic scale-free network. It has been known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying…

Statistical Mechanics · Physics 2007-05-23 Jae Dong Noh , Sang-Woo Kim

We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2 on which this is based. The framework has no need for coin…