Diffusive persistence and the `sign-time' distribution

TL;DR

This paper introduces a novel method using sign-time distribution to accurately determine the persistence exponent for diffusion processes across various dimensions, revealing a significant change in behavior around d ~ 36.

Contribution

The authors derive an exact formula for the persistence exponent in any dimension using a new approach based on sign-time distribution, validated by numerical evidence.

Findings

Exact formula for persistence exponent in arbitrary dimension

Excellent agreement with previous numerical results

Identification of a qualitative change in distribution above d ~ 36

Abstract

We present a new method for extracting the persistence exponent theta for the diffusion equation, based on the distribution P of `sign-times'. With the aid of a numerically verified Ansatz for P we derive an exact formula for theta in arbitrary spatial dimension d. Our results are in excellent agreement with previous numerical studies. Furthermore, our results indicate a qualitative change in P above d ~ 36, signalling the existence of a sharp change in the ergodic properties of the diffusion field.