Global first passage times on fractal lattices

TL;DR

This paper derives explicit formulas for the mean and variance of global first passage times on five fractal lattices, revealing how the variance scales with the number of nodes and the spectral dimension.

Contribution

It provides exact expressions for the first and second moments of first passage times on fractals, linking variance scaling to spectral dimension.

Findings

Variance of first passage time scales as N^(4/d̄)

Explicit formulas for mean first passage time on five fractals

Variance depends on spectral dimension

Abstract

The global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time T. We find simple expressions for the mean global first passage time <T> for five fractals. We also find an exact expression for the second moment <T^2> and show that the variance of the first passage time, Var(T), scales with the number of nodes within the fractal N such that Var(T) ~ N^(4/\bar{d}), where \bar{d} is the spectral dimension.