Statistics of Multiple Sign Changes in a Discrete Non-Markovian Sequence

Abstract

We study analytically the statistics of multiple sign changes in a discrete non-Markovian sequence ,ψ_i=ϕ_i+ϕ_{i-1} (i=1,2....,n) where ϕ_i's are independent and identically distributed random variables each drawn from a symmetric and continuous distribution ρ(ϕ). We show that the probability P_m(n) of m sign changes upto n steps is universal, i.e., independent of the distribution ρ(ϕ). The mean and variance of the number of sign changes are computed exactly for all n>0. We show that the generating function {\tilde P}(p,n)=\sum_{m=0}^{\infty}P_m(n)p^m\sim \exp[-θ_d(p)n] for large n where the `discrete' partial survival exponent θ_d(p) is given by a nontrivial formula, θ_d(p)=\log[{{\sin}^{-1}(\sqrt{1-p^2})}/{\sqrt{1-p^2}}] for 0\le p\le 1. We also show that in the natural scaling limit when m is large, n is large but but keeping x=m/n fixed, P_m(n)\sim \exp[-n Φ(x)] where the large deviation function Φ(x) is computed. The implications of these results for Ising spin glasses are discussed.