Survival-time statistics for sample space reducing stochastic processes

Abstract

Stochastic processes wherein the size of the state space is changing as a function of time offer models for the emergence of scale-invariant features observed in complex systems. I consider such a sample-space reducing (SSR) stochastic process that results in a random sequence of strictly decreasing integers {x(t)},0≤t≤τ, with boundary conditions x(0)=N and x(τ) = 1. This model is shown to be exactly solvable: P_{N}(τ), the probability that the process survives for time τ is analytically evaluated. In the limit of large N, the asymptotic form of this probability distribution is Gaussian, with mean and variance both varying logarithmically with system size: 〈τ〉∼lnN and σ_{τ}^{2}∼lnN. Correspondence can be made between survival-time statistics in the SSR process and record statistics of independent and identically distributed random variables.

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

“…As noted that the SSR process has correspondence with records statistics in iid random variables, and the records statistics, in this case, can be mapped with statistics of cycles in random permutation with uniform measure [21]. Also, polymerization that consists of fragmentation and aggregation processes can be mapped to the formation of cycles in random permutation with uniform measure [17].…”

“…We begin with recalling the definition of SSR stochastic process [15]. This can be visualized as a directed random arXiv:1609.05727v2 [cond-mat.stat-mech] 28 Aug 2017 hopping process on Z + , the set of positive integers [21]. Let us denote the instantaneous state of the process by x(t) = K, where 1 ≤ K ≤ N ; or in terms of a random walk, the walker is at Kth site after time t. The dynamical update rules are the following: At the next time step t+1, the walker can go to any site in the interval [1, K −1] with equal probability, i.e., 1/(K − 1).…”

“…Some interesting examples studied with this point of view are the following: A mapping of percolation on trees and Brownian excursions [19], or the directed abelian sandpile on a narrow strip and a random walk on a ring [20] or the fragmentation and aggregation process in polymerization and the study of cycles in random permutation with uniform measure [17]. In the present context, a mapping between survival time statistics for SSR processes and records statistics of independent and identically distributed (iid) random variables [21] is most pertinent.…”

“…We first review the recent developments on the SSR process; in Ref. [21] the statistics of survival time (the life span) of the SSR process was studied. Exact results, supported by simulations, showed that both mean and variance of the survival time vary in a logarithmic manner as a function of system size, and the asymptotic probability distribution is Gaussian.…”

On multidimensional record patterns

“…In spite of its simplicity and of its efficiency, the latter construction is not frequently met in the literature. Let us however mention that it is the prototypical example of a class of sample-space-reducing (SSR) processes put forward very recently [38,39]. Consider a finite signal of length L.…”