Violation of Scaling in the Contact Process with Quenched Disorder

TL;DR

This paper investigates the two-dimensional contact process with quenched disorder, revealing violations of traditional scaling laws and demonstrating that certain critical exponents are undefined due to anomalous dynamic behavior.

Contribution

It provides the first detailed analysis of the static and dynamic critical exponents in the disordered contact process, showing scaling violations and the breakdown of conventional dynamic scaling.

Findings

Critical exponents beta and nu_perp determined.

Dynamic behavior incompatible with standard scaling.

Logarithmic time dependence observed in simulations.

Abstract

We study the two-dimensional contact process (CP) with quenched disorder (DCP), and determine the static critical exponents beta and nu_perp. The dynamic behavior is incompatible with scaling, as applied to models (such as the pure CP) that have a continuous phase transition to an absorbing state. We find that the survival probability (starting with all sites occupied), for a finite-size system at critical, decays according to a power law, as does the off-critical density autocorrelation function. Thus the critical exponent nu_parallle, which governs the relaxation time, is undefined, since the characteristic relaxation time is itself undefined. The logarithmic time-dependence found in recent simulations of the critical DCP [Moreira and Dickman, Phys. Rev. E54, R3090 (1996)] is further evidence of violation of scaling. A simple argument based on percolation cluster statistics yields a similar logarithmic evolution.