A Model of Directed Walks with Random Self-Interactions

We study a simple one-dimensional model of a folded polymer with random self-interactions. A numerical study of the specific heat shows two regimes: at high temperature, the specific heat looks smooth and sample independent, whereas at low temperature it possesses many narrow peaks which change with the sample considered. The model is simple enough to allow for a full description of its ground states. We obtain numerical evidence for the presence of a “weak freezing” transition and derive an upper bound for the transition temperature. Heuristic arguments provide an estimate of a critical exponent γ(T) which varies continuously with the temperature in the low-temperature phase.