Finite time and asymptotic behaviour of the maximal excursion of a random walk

TL;DR

This paper derives the limit distribution of the maximal excursion of a random walk in any dimension, provides exact early-time behavior results, and explores the diffusion exponent's overshoot and scaling properties.

Contribution

It presents a closed-form limit distribution for the maximal excursion of random walks and analyzes the early-time diffusion exponent behavior across different dimensions and structures.

Findings

Limit distribution obtained in closed form for homogeneous environments.

Exact early-time behavior of the diffusion exponent in one dimension.

Numerical evidence of scaling law in two dimensions.

Abstract

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed form and is an approximation of the exact distribution comparable to that obtained by real space renormalization methods. Then we focus on the early time behaviour of this quantity. The instantaneous diffusion exponent $\nu_n$ exhibits a systematic overshooting of the long time exponent. Exact results are obtained in one dimension up to third order in $n^{-1/2}$. In two dimensions, on a regular lattice and on the Sierpi\'nski gasket we find numerically that the analytic scaling $\nu_n \simeq \nu+A n^{-\nu}$ holds.