Violation of scaling in the contact process with quenched disorder

Abstract

We study the two-dimensional contact process (CP) with quenched disorder (DCP), and determine the static critical exponents $\ensuremath{\beta}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}.$ 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 the critical point, decays according to a power law, as does the off-critical density autocorrelation function. Thus the critical exponent ${\ensuremath{\nu}}_{||},$ 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 [A. G. Moreira and R. Dickman, Phys. Rev. E 54, R3090 (1996)] is further evidence of violation of scaling. A simple argument based on percolation cluster statistics yields a similar logarithmic evolution.