First-passage times in complex scale-invariant media

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 past decade. 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 of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. 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 media, fractals, anomalous diffusion and scale-free networks.

How a finite potential barrier decreases the mean first-passage time

“…On a flat potential landscape significant progress has been achieved in the theory of MFPTs on arbitrary, finite domains [15]. In particular, the role of compact versus noncompact explorations has been revealed in generality [16].…”

Mean first-passage time for random walks on generalized deterministic recursive trees

“…Related with random walks, mean first-passage time (MFPT) -the expected time that a random walker which starts with equal probability at any node will expend to reach a given target-is of interest, as it appears in important real-life first encounter events, which include network routing, reaction-diffusion processes, epidemic spreading, neuron firing, etc. ; see [4][5][6][7].…”

Random walks on deterministic scale-free networks: Exact results

“…Then, these networks often feature infinite dimensionality, and inhomogeneity has been detected in several real systems [5,6,7,8]. One of the most studied structures in the literature are Scale-Free (SF) networks, which show a degree sequence scaling with a characteristic power law [3,9].…”