Open Access
2014 Random walk with long-range constraints
Yinon Spinka, Ron Peled
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Electron. J. Probab. 19: 1-54 (2014). DOI: 10.1214/EJP.v19-3060


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.


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Yinon Spinka. Ron Peled. "Random walk with long-range constraints." Electron. J. Probab. 19 1 - 54, 2014.


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

zbMATH: 1294.82019
MathSciNet: MR3227061
Digital Object Identifier: 10.1214/EJP.v19-3060

Primary: 82B41
Secondary: 05A16 , 60C05 , 60D05 , 60K35 , 82B20 , 82B26

Keywords: Lipschitz function , phase transition , random graph homomorphism , Random walk

Vol.19 • 2014
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