[Paper Review] Directed percolation and directed animals

Deepak Dhar|arXiv (Cornell University)|Mar 22, 2017

Cellular Automata and Applications3 citations

TL;DR

This paper presents a comprehensive introduction to directed percolation and directed animals from a physicist's perspective, establishing their equivalence to Yang-Lee edge singularities and linking them to stochastic processes like the forest-fire model. It demonstrates how local, non-equilibrium dynamics—such as in the Grassberger-Zhang model—can lead to self-organized criticality without fine-tuning, revealing universal critical behavior via probabilistic cellular automata and quantum spin chain analogies.

ABSTRACT

These lectures provide an introduction to the directed percolation and directed animals problems, from a physicist's point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation to undirected first passage percolation problem are described. Equivalence of the $d$-dimensional directed animals problem to $(d-1)$-dimensional Yang-Lee edge-singularity problem is established. Self-organized critical formulation of the percolation problem, which does not involve any fine-tuning of coupling constants to get critical behavior is briefly discussed.

Motivation & Objective

• To introduce directed percolation and directed animals as fundamental models in statistical physics, emphasizing their relevance to stochastic processes and phase transitions.
• To establish the equivalence between d-dimensional directed animals and (d−1)-dimensional Yang-Lee edge singularities, linking them to critical phenomena.
• To demonstrate how local, stochastic evolution rules—such as in the Grassberger-Zhang model—can yield self-organized criticality without fine-tuning of parameters.
• To explore connections between directed percolation and quantum spin chains through the master equation formalism, enabling use of quantum mechanical insights in stochastic systems.

Proposed method

• Formulates directed percolation via a probabilistic cellular automaton model with discrete time, local rules, and state transitions based on neighboring activity (e.g., Domany-Kinzel model).
• Models the forest-fire process as a Markov process with states: green (0), burning (1), and ash (2), where fire spreads stochastically with probability 1−q^r based on r burning neighbors.
• Introduces continuous-time master equation for the system’s probability distribution, mapping it to a Schrödinger-like equation with a Hamiltonian matrix W_CC′.
• Applies the Grassberger-Zhang construction using real-valued variables x(i,t) evolving via local max-min rules, with random inputs η(i,t), to achieve self-organized criticality.
• Uses the transformation y(i,t) = 1 if x(i,t) ≤ p, 0 otherwise, to map the x-process to a directed site percolation process with wetness probability P_DP(p,T).
• Analyzes undirected percolation analogs by defining y(i,t) via min over neighbors of max(y(j,t), x_ij), ensuring non-increasing y-values and convergence to a fixed point related to infinite cluster fraction.

Experimental results

Research questions

• RQ1 How can directed percolation be formulated as a probabilistic cellular automaton with local, parallel update rules?
• RQ2 What is the relationship between directed animals in d dimensions and the (d−1)-dimensional Yang-Lee edge singularity?
• RQ3 Can a local, stochastic dynamical system achieve self-organized criticality without fine-tuning of control parameters?
• RQ4 How does the interface roughness and burst size distribution in interface growth models relate to directed percolation exponents?
• RQ5 Can equilibrium statistical models be endowed with local stochastic dynamics to naturally reach their critical point at large times?

Key findings

• The d-dimensional directed animals problem is mathematically equivalent to the (d−1)-dimensional Yang-Lee edge singularity problem, providing a deep connection between combinatorics and critical phenomena.
• The Grassberger-Zhang model achieves self-organized criticality via local, stochastic rules: the distribution of x(i,t) converges to the directed percolation wetness probability P_DP(p,T) at time T.
• In the self-organized directed percolation model, the interface roughens with a nontrivial roughness exponent, and burst size distributions exhibit power-law tails, characteristic of critical systems.
• For undirected percolation, a similar local stochastic rule on bond strengths x_ij leads to a fixed-point y*(i) whose survival probability Prob(y* ≥ z) equals the infinite cluster fraction at bond concentration z.
• The master equation for the system’s probability distribution can be mapped to a Schrödinger equation with a Hamiltonian matrix W_CC′, enabling use of quantum spin chain techniques in stochastic dynamics.
• The forest-fire model with regrowth exhibits a non-trivial phase transition even in 1D, with precise critical exponents known numerically, though no exact solution exists yet.