Randomness in vertex models and directed walks

Abstract

The authors consider a d-dimensional random five vertex (modified KDP) model where the vertex energies are site dependent, uncorrelated random numbers (>0). This model maps onto many directed walks in a random environment. They show that the upper critical dimension of the random vertex model is 2. They obtain a bound v>or=3/(d+3) for the size exponent of a directed walk in a random medium. The breakdown of hyperscaling in the vertex model is connected to the anomalous growth of the free energy with an exponent consistent with the corresponding one ( chi =2-1/v) for a single directed walk.