Occupation times of random walks in confined geometries: From random trap model to diffusion limited reactions

TL;DR

This paper analytically derives the exact distribution of occupation times for random walks in confined geometries, with implications for trap models and diffusion-limited reactions, revealing complex dependencies on geometry and parameters.

Contribution

It provides explicit analytical expressions for occupation time distributions in confined domains, connecting random walk theory to trap models and reaction probabilities.

Findings

Exact occupation time distribution in parallelepipedic domains

Non-trivial dependence of mean first passage time on positions

Explicit calculation of reaction probability with imperfect centers

Abstract

We consider a random walk in confined geometry, starting from a site and eventually reaching a target site. We calculate analytically the distribution of the occupation time on a third site, before reaching the target site. The obtained distribution is exact, and completely explicit in the case or parallepipedic confining domains. We discuss implications of these results in two different fields: The mean first passage time for the random trap model is computed in dimensions greater than 1, and is shown to display a non-trivial dependence with the source and target positions ; The probability of reaction with a given imperfect center before being trapped by another one is also explicitly calculated, revealing a complex dependence both in geometrical and chemical parameters.