Spontaneous symmetry breaking in amnestically induced persistence

TL;DR

This paper explores a non-Markovian random walk model with amnestically induced persistence, revealing a complex phase diagram with four distinct phases, including log-periodic behaviors and symmetry properties.

Contribution

It provides the first detailed analysis of phase transitions and symmetry breaking in a non-Markovian walk with amnestically induced persistence, combining numerical and analytical methods.

Findings

Identified four distinct phases in the model's phase diagram.

Discovered log-periodic nonpersistence and persistence with discrete scale invariance.

Provided evidence of geometric self-similarity in log-periodic persistence.

Abstract

We investigate a recently proposed non-Markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence. We report numerical and analytical results showing the complete phase diagram, consisting of 4 phases, for this system: (i) classical nonpersistence, (ii) classical persistence (iii) log-periodic nonpersistence and (iv) log-periodic persistence driven by negative feedback. The first two phases possess continuous scale invariance symmetry, however log-periodicity breaks this symmetry. Instead, log-periodic motion satisfies discrete scale invariance symmetry, with complex rather than real fractal dimensions. We find for log-periodic persistence evidence not only of statistical but also of geometric self-similarity.