Survival Probabilities of History-Dependent Random Walks

TL;DR

This paper studies the survival probabilities of long-memory random walks with an absorbing boundary, revealing a phase transition at a critical correlation strength that affects long-term survival behavior.

Contribution

It introduces an analytically solvable model for history-dependent random walks and identifies a dynamical phase transition based on correlation strength.

Findings

Survival probability is finite for strong positive correlations.

Survival probability decays as a power-law below the critical correlation.

A phase transition occurs at a critical correlation value .

Abstract

We analyze the dynamics of random walks with long-term memory (binary chains with long-range correlations) in the presence of an absorbing boundary. An analytically solvable model is presented, in which a dynamical phase-transition occurs when the correlation strength parameter \mu reaches a critical value \mu_c. For strong positive correlations, \mu > \mu_c, the survival probability is asymptotically finite, whereas for \mu < \mu_c it decays as a power-law in time (chain length).