PMID: 11909139Mar 23, 2002Paper

Statistics of multiple sign changes in a discrete non-Markovian sequence

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Satya N Majumdar

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.

References

Apr 17, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·T W Burkhardt
Jul 20, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·S N MajumdarG C Ehrhardt
Nov 3, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·S N Majumdar, D Dhar

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Citations

Mar 5, 2004·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·G C M A EhrhardtAlan J Bray
May 16, 2018·Physical Review. E·Markus Nyberg, Ludvig Lizana
May 20, 2018·Physical Review. E·Markus NybergTobias Ambjörnsson
Feb 7, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Satya N Majumdar, Olivier C Martin
Nov 22, 2002·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Satya N Majumdar, David S Dean

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