Mean number of encounters of N random walkers and intersection of strongly anisotropic fractals

TL;DR

This paper analyzes the mean encounters of multiple random walkers in a subspace and explores the intersection properties of strongly anisotropic fractals to understand their long-term behavior.

Contribution

It introduces a continuum approximation and Monte Carlo simulations to evaluate encounters and develops a framework for intersecting anisotropic fractals.

Findings

Analytical expression for mean encounters E_N(t)

Numerical validation in 1D and 2D

Long-time behavior characterized through fractal intersections

Abstract

We study the mean number of encounters up to time t, E_N(t), taking place in a subspace with dimension d* of a d-dimensional lattice, for N independent random walkers starting simultaneously from the same origin. E_N is first evaluated analytically in a continuum approximation and numerically through Monte Carlo simulations in one and two dimensions. Then we introduce the notion of the intersection of strongly anisotropic fractals and use it to calculate the long-time behaviour of E_N.