Electronic Journal of Probability

Random walk with long-range constraints

Yinon Spinka and Ron Peled

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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 $\mathbb{Z}$, where the graph $P_{n,d}$ is the discrete segment $\{0,1,\ldots, n\}$ 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.<br /><br />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.<br /><br />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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 52, 54 pp.

Accepted: 23 June 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60C05: Combinatorial probability 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05A16: Asymptotic enumeration 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random walk random graph homomorphism phase transition Lipschitz function

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Spinka, Yinon; Peled, Ron. Random walk with long-range constraints. Electron. J. Probab. 19 (2014), paper no. 52, 54 pp. doi:10.1214/EJP.v19-3060. https://projecteuclid.org/euclid.ejp/1465065694

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