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 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 $\pi(n)$ 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 non-stationary phase to a stationary phase as $c$ increases through a critical value $c_{\rm crit}$. Focusing on $f(x)=e^{-x}$ (with $x\ge 0$), we show that $c_{\rm crit}=1$ and for $c<1$, the record statistics is non-stationary. However, for $c>1$, the record statistics is stationary with avalanche size distribution $\pi(n)\sim n^{-1-\lambda(c)}$ for large $n$. Consequently, for $c>1$, the mean number of records up to $N$ steps grows algebraically $\sim N^{\lambda(c)}$ for large $N$. Remarkably, the exponent $\lambda(c)$ depends continously 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.