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 Pn,d to the integersZ, where the graph Pn,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 Pn,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 logn. We establish this conjecture with the precise threshold log2 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− log2 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.