Phase-Transition in Binary Sequences with Long-Range Correlations

TL;DR

This paper investigates how long-range correlations in binary sequences cause a phase transition from normal to super-diffusive behavior, with implications for analyzing complex data like DNA, texts, and financial time series.

Contribution

It introduces a model showing a phase transition in diffusion behavior due to long-range correlations in binary sequences, extending understanding of correlated stochastic processes.

Findings

System undergoes a phase transition from normal to super-diffusion.

The critical correlation strength determines the transition point.

Results are applicable to DNA, texts, and financial data.

Abstract

Motivated by novel results in the theory of correlated sequences, we analyze the dynamics of random walks with long-term memory (binary chains with long-range correlations). In our model, the probability for a unit bit in a binary string depends on the fraction of unities preceding it. We show that the system undergoes a dynamical phase-transition from normal diffusion, in which the variance D_L scales as the string's length L, into a super-diffusion phase (D_L ~ L^{1+|alpha|}), when the correlation strength exceeds a critical value. We demonstrate the generality of our results with respect to alternative models, and discuss their applicability to various data, such as coarse-grained DNA sequences, written texts, and financial data.