Statistical properties of avalanches via the c-record process

Abstract

We study the statistics of avalanches, as a response to an applied force, undergone by a particle hopping on a one dimensional lattice where the pinning forces at each site are independent and identically distributed (I.I.D), each drawn from a continuous f (x). The avalanches in this model correspond to the inter-record intervals in a modified record process of I.I.D variables, defined by a single parameter c > 0. This parameter characterizes the record formation via the recursive process R k > R k−1 −c, where R k denotes the value of the k-th record. We show that for c > 0, if f (x) decays slower than an exponential for large x, the record process is nonstationary as in the standard c = 0 case. In contrast, if f (x) has a faster than exponential tail, the record process becomes stationary and the avalanche size distribution π(n) has a decay faster than 1/n 2 for large n. The marginal case where f (x) decays exponentially for large x exhibits a phase transition from a non-stationary phase to a stationary phase as c increases through a critical value ccrit. Focusing on f (x) = e −x (with x ≥ 0), we show that ccrit = 1 and for c < 1, the record statistics is non-stationary. However, for c > 1, the record statistics is stationary with avalanche size distribution π(n) ∼ n −1−λ(c) for large n. Consequently, for c > 1, the mean number of records up to N steps grows algebraically ∼ N λ(c) for large N . Remarkably, the exponent λ(c) depends continously on c for c > 1 and is given by the unique positive root of c = − ln(1 − λ)/λ. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.

Record statistics for random walks and Lévy flights with resetting

“…When the entries are random variables (either independent or correlated), R N is also a random variable and studying its statistics is what is called 'record statistics'. The study of record statistics started in the statistics and probability literature [25][26][27][28][29], but off late it has generated much interest in the statistical physics literature also, in various contexts such as in biological evolution models [30][31][32][33][34], random walks (RWs) [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], avalanches in disordered systems [20,36,52] etc. From the statistical physics point of view, one of the central issues is the universality of the statistics of R N with respect to the underlying probability distribution of the entries [24].…”