Percolation, sliding, localization and relaxation in topologically closed circuits

TL;DR

This paper investigates the relaxation dynamics and spectral properties of random walks in topologically closed circuits, revealing thresholds for under-damped relaxation and phenomena like complexity saturation as bias increases.

Contribution

It introduces new insights into the spectral behavior and relaxation modes of stochastic processes in closed circuits, highlighting differences from non-hermitian Hamiltonian models.

Findings

Identifies the threshold for under-damped relaxation.

Observes complexity saturation with increased bias.

Analyzes implications of percolation and sliding transitions.

Abstract

Considering a "random walk in a random environment" in a topologically closed circuit, we explore the implications of the percolation and sliding transitions for its relaxation modes. A complementary question regarding the "delocalization" of eigenstates of non-hermitian Hamiltonians has been addressed by Hatano, Nelson, and followers. But we show that for a conservative stochastic process the implied spectral properties are dramatically different. In particular we determine the threshold for under-damped relaxation, and observe "complexity saturation" as the bias is increased.