Moments and distribution of the local times of a transient random walk on \Z^d

2007/08/31 by Mathias Becker, Wolfgang König, Becker, Mathias +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #60F15 #60G50 #60J55 #Diffusion and Search Dynamics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0708.4408

openalex publication_date 2007/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider an arbitrary transient random walk on \Z^d with d∈\N. Pick α∈[0,∞) and let L_n(α) be the spatial sum of the α-th power of the n-step local times of the walk. Hence, L_n(0) is the range, L_n(1)=n+1, and for integers α, L_n(α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L_n(α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý \citeCe07. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.

Related

• Exponential moments of self-intersection local times of stable random walks in subcritical dimensions
• Strong law of large numbers for a function of the local times of a transient random walk in \mathbb Z^d
• Optimal bounds for self-intersection local times
• Strong law of large numbers for a function of the local times of a transient random walk on groups
• Upper tails of self-intersection local times of random walks: survey of proof techniques