Persistence exponent of the diffusion equation in epsilon dimensions

TL;DR

This paper derives a small-d expansion for the persistence exponent of the diffusion equation in epsilon dimensions, revealing how the persistence behavior varies with dimension.

Contribution

It introduces a perturbation theory approach to compute the persistence exponent for the diffusion equation in low dimensions.

Findings

Derived an explicit small-d expansion for the persistence exponent

Identified the Gaussian Markovian limit as d approaches zero

Provided numerical estimates for the correction terms

Abstract

We consider the d-dimensional diffusion equation for a field phi(x,t) with random initial condition, and observe that, when appropriately scaled, phi(0,t) is Gaussian and Markovian in the limit d->0. This leads via the Majumdar-Sire perturbation theory to a small-d expansion for the persistence exponent theta(d). We find theta(d) = d/4 - 0.12065...d^{3/2} + ...