First-passage times in complex scale-invariant media

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 past decade1. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media2, 3, neuron firing dynamics4, spreading of diseases5 or target search processes6, 7, 8, 9. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media1. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source–target distance. The 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 representative models of disordered media10, fractals3, anomalous diffusion11 and scale-free networks12.