通过c记录过程的雪崩的统计特性
Statistical properties of avalanches via the c-record process
摘要：
我们研究了雪崩的统计，作为一个响应所施加的力，经历了一个粒子跳跃的一维晶格中的钉扎力在每个网站是独立的和相同的分布（I.I.D），每个绘制从一个连续的$f(x)$。在这个模型中的雪崩对应于I.I.D变量的修改记录过程中的记录间间隔，由单个参数$c>0$定义。该参数通过递归过程$R_k > R_{k-1}-c$表征记录形成，其中$R_k$表示第$k$条记录的值。我们发现，对于$c>0$，如果$f(x)$衰减慢于大$x$的指数，记录过程是非平稳的标准$c=0$的情况下。相比之下，如果$f(x)$具有比指数更快的尾部，则记录过程变得稳定，并且对于大的$n$，雪崩尺寸分布$π(n)$具有比$1/n^2$更快的衰减。当$c$通过临界值$c_{\rm crit}$增加时，对于大的$x$，$f(x)$指数衰减的边缘情况表现出从非稳定相到稳定相的相变。专注于$f(x)=e^{-x}$（与$x\ge 0$），我们表明，$c_{\rm crit}=1$和$c<1$，记录统计是非平稳的。然而，对于$c>1$，记录统计数据是稳定的，对于大型$n$，雪崩尺寸分布为$π(n)\sim n^{-1-λ(c)}$。因此，对于$c>1$，对于大的$N$，直到$N$步骤的平均记录数代数地增长$\sim N^{λ(c)}$。值得注意的是，对于$c>1$，指数$λ(c)$连续依赖于$c$，并且由$c=-\ln (1-λ)/λ$的唯一正根给出。我们还揭示了稳定阶段类似地震序列的雪崩之间的非平凡相关性的存在。
英文摘要：
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 $c_{\rm crit}$. 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 $π(n)\sim n^{-1-λ(c)}$ for large $n$. Consequently, for $c>1$, the mean number of records up to $N$ steps grows algebraically $\sim N^{λ(c)}$ 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.
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