Local Reversibility and Divergent Markov Length in 1+1-D Directed Percolation

TL;DR

This paper explores the concept of local reversibility and Markov length in the 1+1-D directed percolation model, revealing divergence at critical points and demonstrating the utility of information-theoretic measures in identifying phase transitions.

Contribution

It introduces the application of local reversibility and Markov length analysis to classical non-equilibrium models, showing divergence at criticality and the effectiveness of conditional mutual information as a phase transition indicator.

Findings

Markov length diverges at the directed percolation critical point.

Conditional mutual information detects phase transitions despite diverging Markov length.

Active phase exhibits local reversibility, contrasting with classical equilibrium behavior.

Abstract

Recent progress in open many-body quantum systems has highlighted the importance of the Markov length, the characteristic scale over which conditional correlations decay. It has been proposed that non-equilibrium phases of matter can be defined as equivalence classes of states connected by short-time evolution while maintaining a finite Markov length, a notion called local reversibility. A natural question is whether well-known classical models of non-equilibrium criticality fit within this framework. Here we investigate the Domany-Kinzel model -- which exhibits an active phase and an absorbing phase separated by a 1+1-D directed-percolation transition -- from this information-theoretic perspective. Using tensor network simulations, we provide evidence for local reversibility within the active phase. Notably, the Markov length diverges upon approaching the critical point, unlike classical equilibrium transitions where Markov length is zero due to their Gibbs character. Correspondingly, the conditional mutual information exhibits scaling consistent with directed percolation universality. Further, we analytically study the case of 1+1-D compact directed percolation, where the Markov length diverges throughout the phase diagram due to spontaneous breaking of domain-wall parity symmetry from strong to weak. Nevertheless, the conditional mutual information continues to faithfully detect the corresponding phase transition.