1. Université Pierre et Marie Curie-Paris 6, Laboratoire de Physique Théorique de la Matière Condensée, UMR CNRS 7600, case 121, 4 Place Jussieu, 75005 Paris, France
2. School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel

Correspondence to: O. Bénichou¹ Correspondence and requests for materials should be addressed to O.B. (Email: benichou@lptmc.jussieu.fr).

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 decade¹. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media^{2, 3}, neuron firing dynamics⁴, spreading of diseases⁵ or target search processes^{6, 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.

1. Université Pierre et Marie Curie-Paris 6, Laboratoire de Physique Théorique de la Matière Condensée, UMR CNRS 7600, case 121, 4 Place Jussieu, 75005 Paris, France
2. School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel

Correspondence to: O. Bénichou¹ Correspondence and requests for materials should be addressed to O.B. (Email: benichou@lptmc.jussieu.fr).

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