Boundary of the Range of a random walk and the Fölner property

2018/10/24 by George Deligiannidis, Deligiannidis, George, Sébastien Gouëzel +3
Mathematics · #28D99 #60F15 #60G50 #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1810.10454

openalex publication_date 2018/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The range process R_n of a random walk is the collection of sites visited by the random walk up to time n. In this work we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Fölner sequence and show the following results: % (a) The size of the inner boundary |∂ R_n| of the range of recurrent aperiodic random walks on ℤ² with finite variance and aperiodic random walks in ℤ in the standard domain of attraction of the Cauchy distribution, divided by (n)/(log²(n)), converges to a constant almost surely. % (b) We establish a formula for the Fölner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for transient random walk on groups which are not virtually cyclic, for almost every path, the range is not a Fölner sequence. % (c) For aperiodic random walks in the domain of attraction of symmetric α- stable distributions with 1

Citations

• Recurrence of Co-Cycles and Random Walks

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