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

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 nonequilibrium 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 nonequilibrium 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. The power-law decay of conditional correlations sharply quantifies the non-Gibbs nature of the directed-percolation critical point and also introduces a new critical exponent whose value we estimate using our simulations. 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.