First-passage times in complex scale-invariant media

TL;DR

This paper develops a universal theoretical framework to accurately evaluate mean first passage times in complex, scale-invariant media, applicable to various stochastic processes and confirmed by numerical simulations.

Contribution

It introduces a general analytical approach for calculating mean FPTs in complex media, extending beyond traditional 1D or homogeneous models.

Findings

Provides a universal scaling law for MFPT based on domain volume and source-target distance.

Validates the theory with numerical simulations across different models.

Applicable to disordered media, fractals, anomalous diffusion, and scale-free networks.

Abstract

How long does it take a random walker to reach a given target point? This quantity, known as a first passage time (FPT), has led to a growing number of theoretical investigations over the last decade1. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods to determine the FPT properties in confining domains have been limited to effective 1D geometries, or for space dimensions larger than one only to homogeneous media1. Here we propose a general theory which allows one to accurately evaluate the mean FPT (MFPT) in complex media. Remarkably, this analytical approach provides a universal scaling dependence of the MFPT on both the volume of the confining domain and the source-target distance. This analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties. Our theoretical predictions are confirmed by numerical simulations for several emblematic models of disordered media, fractals, anomalous diffusion and scale free networks.