Dynamics as a substitute for replicas in systems with quenched random impurities

Abstract

Traces over random degrees of freedom may be performed ab initio, without replicas, provided one is using a dynamic description of the system. An effective Lagrangian is obtained after the random degrees of freedom are eliminated. An order parameter is then defined, à la Edwards and Anderson, in terms of the time-persistent part of the correlation function. Taking a system with random temperature as a model example, we characterize to all orders the equation of state in the ordered phase. It is found that this equation contains odd powers of the order parameter that destabilize the simplest solutions. These terms suggest that the $t^{-\frac{1}{2}}$ decay law, found by Ma and Rudnick for the correlation function, is restricted to a region close to the presumed transition point. The formalism is particularly suitable for treating dynamics in the ordered phase.