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

Area of Science:

• Statistical Mechanics
• Probability Theory
• Condensed Matter Physics

Background:

• Non-Markovian sequences exhibit complex behavior.
• Understanding sign change statistics is crucial for analyzing dynamic systems.
• Previous studies often focused on Markovian processes.

Purpose of the Study:

• To analytically investigate the statistics of multiple sign changes in discrete non-Markovian sequences.
• To determine the universality and derive exact statistical properties of sign changes.
• To explore the large deviation behavior and its connection to physical systems.

Main Methods:

• Analytical study of a discrete non-Markovian sequence defined as psi(i)=phi(i)+phi(i-1).
• Independent and identically distributed random variables phi(i) drawn from symmetric, continuous distributions.
• Calculation of probability, mean, variance, and generating functions for sign changes.

Main Results:

• The probability of m sign changes, P(m)(n), is universal and independent of the underlying distribution.
• Exact formulas for the mean and variance of sign changes are derived.
• Asymptotic analysis reveals approximate exponential decay for the generating function and a large deviation function Phi(x).

Conclusions:

• The universality of sign change probability simplifies analysis across different distributions.
• The derived formulas offer precise quantitative predictions for the number of sign changes.
• The findings have potential implications for understanding phenomena in Ising spin glasses.