Random Walk with Long-Range Constraints

Abstract: 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,d} to the integers Z, where the graph P_{n,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,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.