Critical Dynamics of the Contact Process with Quenched Disorder

TL;DR

This paper investigates how quenched disorder affects the critical spreading dynamics of the two-dimensional contact process, revealing altered critical exponents and supporting the single-phase transition conjecture in disordered systems.

Contribution

It provides the first detailed analysis of critical exponents in the disordered 2D contact process, showing significant deviations from the pure model and confirming the single transition hypothesis.

Findings

Disorder changes critical exponents significantly.

Hyperscaling relation is violated in the disordered model.

Supports the conjecture of a single phase transition in disordered systems.

Abstract

We study critical spreading dynamics in the two-dimensional contact process (CP) with quenched disorder in the form of random dilution. In the pure model, spreading from a single particle at the critical point $\lambda_c$ is characterized by the critical exponents of directed percolation: in $2+1$ dimensions, $\delta = 0.46$, $\eta = 0.214$, and $z = 1.13$. Disorder causes a dramatic change in the critical exponents, to $\delta \simeq 0.60$, $\eta \simeq -0.42$, and $z \simeq 0.24$. These exponents govern spreading following a long crossover period. The usual hyperscaling relation, $4 \delta + 2 \eta = d z$, is violated. Our results support the conjecture by Bramson, Durrett, and Schonmann [Ann. Prob. {\bf 19}, 960 (1991)], that in two or more dimensions the disordered CP has only a single phase transition.