TY - JOUR

T1 - Random walk with long-range constraints

AU - Peled, Ron

AU - Spinka, Yinon

PY - 2014

Y1 - 2014

N2 - 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 integers ℤ, 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 log n. 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.

AB - 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 integers ℤ, 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 log n. 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.

KW - Constrained random walk

KW - Homomorphism height function

KW - Localization/delocalization transition

KW - Random graph homomorphism

UR - http://www.scopus.com/inward/record.url?scp=84903272750&partnerID=8YFLogxK

U2 - 10.1214/EJP.v19-3060

DO - 10.1214/EJP.v19-3060

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AN - SCOPUS:84903272750

SN - 1083-6489

VL - 19

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 52

ER -