Random walk with long-range constraints

We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly sampled from all graph homomorphisms from the graph P-n,P-d to the integers Z, where the graph P-n,P-d is the discrete segment {0, 1, . . . , n} with edges between vertices of different parity whose distance is at most 2d | 1. Such a graph homomorphism can be viewed as a height function whose values change by exactly one along edges of the graph P-n,P-d. We also consider a similarly defined model on the discrete torus. Benjamini, Yadin and Yehudayoff conjectured that this model undergoes a phase transition from a delocalized to a localized phase when d grows beyond a threshold c log n. We establish this conjecture with the precise threshold log(2) n. Our results provide information on the typical range and variance of the height function for every given pair of n and d, including the critical case when d - log(2) n tends to a constant. In addition, we identify the local limit of the model, when d is constant and n tends to infinity, as an explicitly defined Markov chain.