Correspondence between noisy sample space reducing process and records in correlated random events

Abstract

We study survival time statistics in a noisy sample space reducing (SSR) process. Our simulations suggest that both the mean and standard deviation scale as $\sim N/N^{\lambda}$, where $N$ is the system size and $\lambda$ is a tunable parameter that characterizes the process. The survival time distribution has the form $\mathcal{P}_{N}(\tau)\sim N^{-\theta}J(\tau/N^{\theta})$, where $J$ is a universal scaling function and $\theta = 1-\lambda$. Analytical insight is provided by a conjecture for the equivalence between the survival time statistics in the noisy SSR process and the record statistics in a correlated time series modeled as drifted random walk with Cauchy distributed jumps.