Statistical properties of avalanches via the c-record process

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 interrecord 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 > Rk-1 - c, where R-k denotes the value of the kth 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 pi(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 nonstationary phase to a stationary phase as c increases through a critical value c(crit). Focusing on f (x) = e(-x) (with x >= 0), we show that c(crit) = 1 and for c < 1, the record statistics is nonstationary. However, for c > 1, the record statistics is stationary with avalanche size distribution pi(n) similar to n(-1-lambda(c)) for large n. Consequently, for c > 1, the mean number of records up to N steps grows algebraically similar to N-lambda(c) for large N. Remarkably, the exponent lambda(c) depends continuously on c for c > 1 and is given by the unique positive root of c = - ln(1 - lambda)/lambda. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.