RANDOMNESS IN VERTEX MODELS AND DIRECTED WALKS

We 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. We show that the upper critical dimension of the random vertex model is 2. We obtain a bound nu greater-than-or-equal-to 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/nu) for a single directed walk.