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 psi(i)=phi(i)+ phi(i-1) (i=1,2, em leader,n) where phi(i)'s are independent and identically distributed random variables each drawn from a symmetric and continuous distribution rho(phi). We show that the probability P(m)(n) of m sign changes up to n steps is universal, i.e., independent of the distribution rho(phi). 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)= summation operator(infinity)(m=0)P(m)(n)p(m) approximately exp[-theta(d)(p)n] for large n where the "discrete" partial survival exponent theta(d)(p) is given by a nontrivial formula, theta(d)(p)=ln[sin(-1)(square root of [1-p(2)])/square root of [1-p(2)]] for 0< or = p < or = 1. We also show that in the natural scaling limit m-->infinity, n-->infinity but keeping x=m/n fixed, P(m)(n) approximately exp[-n Phi (x)] where the large deviation function Phi(x) is computed. The implications of these results for Ising spin glasses are discussed.