1+1-d Directed Percolation Demonstrates Local Reversibility and Finite Markov Length in Active Phases

The behaviour of complex systems far from equilibrium remains a fundamental challenge in physics, and recent work highlights the importance of the Markov length, a measure of how quickly correlations fade over time. Yu-Hsueh Chen and Tarun Grover, both from the University of California at San Diego, alongside their colleagues, investigate whether this concept of ‘local reversibility’, defining phases by stable short-term evolution and finite Markov length, applies to established models of non-equilibrium criticality. They focus on the Domany-Kinzel model, a well-known system undergoing a transition between active and absorbing states, and demonstrate local reversibility within the active phase using advanced tensor network simulations. Significantly, the team finds that the Markov length diverges as the system approaches its critical point, a behaviour distinct from traditional equilibrium transitions, and this divergence accurately reflects the underlying physics of directed percolation.

Mutual Information Reveals Correlation Efficiency

Researchers are investigating the behaviour of complex systems far from equilibrium, focusing on the concept of the Markov length, which measures how quickly correlations fade over time. This work explores whether ‘local reversibility’, defined by stable short-term evolution and a finite Markov length, applies to established models of non-equilibrium criticality. The team focused on the Domany-Kinzel model, a system undergoing a transition between active and absorbing states, and demonstrated local reversibility within its active phase using advanced tensor network simulations.

Significantly, the team discovered that the Markov length diverges as the system approaches its critical point, a behaviour unlike traditional equilibrium transitions. This divergence accurately reflects the underlying physics of directed percolation, a process where a signal propagates through a disordered medium. To quantify this behaviour, scientists analysed how information is distributed within the system using a measure called conditional mutual information, which reveals the correlation between different regions of the system.

The analysis revealed that this conditional mutual information decays exponentially with distance, allowing the team to define the Markov length as the characteristic distance over which this decay occurs. Experiments involved simulating the system’s evolution from an active state, tracking how quickly it relaxed to a stable state, and measuring the resulting Markov length.

Markov Length Diverges Near Criticality

This study investigates non-equilibrium phases of matter by examining the Domany-Kinzel model and directed percolation, focusing on the concept of local reversibility and the Markov length. Researchers employed tensor network simulations to analyse the Domany-Kinzel model, providing evidence for local reversibility within its active phase. Crucially, the team demonstrated that the Markov length diverges as the system approaches its critical point, a stark contrast to equilibrium transitions where the Markov length is zero.

To quantify this behaviour, scientists defined the Markov length through the conditional mutual information, which measures the correlation between different regions of the system. The team observed that this conditional mutual information decays exponentially with the size of a buffer region, allowing them to define the Markov length as the characteristic distance over which this decay occurs. Experiments involved simulating the time evolution of the system starting from an active state, tracking how quickly it relaxed to a stationary state, and measuring the resulting Markov length.

Further analysis focused on 1+1-dimensional compact directed percolation, where researchers analytically showed that the Markov length diverges throughout the entire phase diagram due to a spontaneous breaking of symmetry. Despite this divergence, the conditional mutual information continued to accurately detect the phase transition, confirming the robustness of this approach.

Markov Length Defines Active Phase Behaviour

Recent work highlights the importance of the Markov length, a measure of how quickly correlations decay within a system, in defining non-equilibrium phases of matter. Researchers have proposed that states connected by short-time evolution with a finite Markov length belong to the same phase, a concept termed local reversibility. This study investigates the Domany-Kinzel model, a classical system exhibiting a transition between active and absorbing phases, to determine if it aligns with this framework.

Using tensor network simulations, scientists provide evidence for local reversibility within the active phase of the model. Notably, the Markov length diverges as the system approaches the critical point, a behaviour distinct from equilibrium transitions where the Markov length remains zero. This divergence indicates a fundamentally different behaviour, suggesting the critical point is strongly non-Gibbsian. The conditional mutual information exhibits scaling consistent with directed percolation universality, further supporting these findings.

Further analysis of 1+1-dimensional compact directed percolation reveals that the Markov length diverges throughout the entire active phase, due to a spontaneous breaking of symmetry. Despite this divergence, the conditional mutual information continues to accurately identify the phase transition, demonstrating its robustness as a diagnostic tool. Measurements confirm that the Markov length scales with system size, indicating a genuine divergence.

Domany-Kinzel Model Exhibits Diverging Markov Length

This research investigates the Domany-Kinzel model, a well-known example of a non-equilibrium system, through the lens of local reversibility and the Markov length, a measure of how quickly correlations decay within the system. The team demonstrates evidence for local reversibility within the active phase of the model, meaning states can evolve into one another via short-depth processes.

Importantly, the Markov length diverges as the system approaches its critical point, a behaviour distinct from equilibrium transitions where the Markov length remains zero. This divergence correlates with scaling consistent with directed percolation universality. Further analysis extended to the case of 1+1-dimensional compact directed percolation, revealing a divergent Markov length throughout the entire phase diagram due to a spontaneous breaking of symmetry.

Despite this divergence, the conditional mutual information continues to accurately identify the phase transition. The findings support the idea that a finite Markov length is central to defining non-equilibrium phases of matter and provide a new perspective on understanding critical phenomena in systems with absorbing states.