The random walk penalised by its range in dimensions d≥ 3

2018/11/12 by Nathanaël Berestycki, Berestycki, Nathanael, Raphaël Cerf +1 · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1811.04700

openalex publication_date 2018/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a self-attractive random walk such that each trajectory of length N is penalised by a factor proportional to exp ( - |R_N|), where R_N is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately ρ_d N^1/(d+2), for some explicit constant ρ_d >0. This proves a conjecture of Bolthausen who obtained this result in the case d=2.

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