Low-temperature properties of directed walks with random self interactions

We consider a simple model for a chain with random self-interactions. Each monomer on the chain carries a charge, which is a random variable, and the configurations of the chain are directed walks. We show that at zero temperature the end-to-end distance R and the energy E of chains with total charge Q approximately Nx scale as R approximately Nnu (x) and E approximately Ntheta (x). The exponents thus depend on the total charge through the parameter x. These exponents are measured numerically and compared to the predictions of some simple arguments. At a finite temperature, R is extensive and appears to be independent of x.