Emergence of power law distributions for odd and even lifetimes

Abstract

Avalanche lifetime distributions have been related to first-return random walk processes. In this sense, the theory for random walks can be employed to understand, for instance, the origin of power law distributions in self-organized criticality. In this work, we study first-return probability distributions, f(n), for discrete random walks with constant one-step transition probabilities. Explicit expressions are given in terms of 2F1 hypergeometric functions, allowing us to study the different behaviors of f(n) for odd and even values of n. We show that the first-return probabilities have a power law behavior with exponent -3/2 only when the random walk is unbiased. In any other case, it presents an exponential decay.