On self-attracting \(d\)-dimensional random walks
type: Item
instance of: scholarly article
title: On self-attracting \(d\)-dimensional random walks
zbMATH Open document ID: 0873.60008
author: Erwin Bolthausen
author: Uwe Schmock
published in: The Annals of Probability
publication date: 1997-10-16T00:00:00Z
review text: Let \(\{X_t\}_{t\geq 0}\) be a symmetric, nearest-neighbour random walk on \(\mathbb{Z}^d\) with exponential holding times of expectation \(1/d\), starting at the origin. For a potential \(V : \mathbb{Z}^d\to [0,\infty)\) with finite and nonempty support, transformed path measures \(\widehat P_T\), \(T>0\), are suitably defined. The authors prove that if \(d=1\) or if the self-attraction is sufficiently strong, then \(|X_t|_\infty\) has an exponential moment under \(\widehat P_t\) which is uniformly bounded for \(T>0\) and \(t\in [0,T]\). The main result is that \(\{X_t\}_{t\geq 0}\) under suitable subsequences of \(\{\widehat P_T\}_{T>0}\) behaves for large \(T\) asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, the authors even prove convergence of the transformed path measures and the law of \(X_T\) as well as the law of the empirical measure \(L_T\) under \(\{\widehat P_T\}_{T>0}\). The Donsker-Varadhan large deviation theory plays a crucial role in the authors' approach.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1018401
zbMATH Keywords: nearest-neighbour random walk
zbMATH Keywords: self-attraction
zbMATH Keywords: strong Dirac-type interaction
zbMATH Keywords: path measures
zbMATH Keywords: large deviation theory
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1024404411
OpenAlex ID: W2084434435
cites work: Localization of a two-dimensional random walk with an attractive path interaction
cites work: Convergence of path measures arising from a mean field or polaron type interaction
cites work: The diffusive phase of a model of self-interacting walks
cites work: Q4003728
cites work: Reinforced random walk
cites work: Invariance principle and empirical mean large deviations of the critical Ornstein-Uhlenbeck process
cites work: Asymptotic evaluation of certain markov process expectations for large time, II
cites work: Asymptotic evaluation of certain Markov process expectations for large time—III
cites work: Q3721531
cites work: Q4693177
cites work: The free energy of the dirac polaron, an explicit solution
cites work: General Irreducible Markov Chains and Non-Negative Operators
cites work: Phase transition in reinforced random walk and RWRE on trees
cites work: Q5533878
cites work: Q3791946
cites work: On the confinement property of two‐dimensional Brownian motion among poissonian obstacles
DOI: 10.1214/AOP/1024404411
description: scientific article; zbMATH DE number 1018401