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**Random walk with long-range constraints.**
*(English)*
Zbl 1294.82019

Summary: We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by I. Benjamini et al. [Electron. J. Probab. 12, 926–950 (2007; Zbl 1127.60007)] 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 \(\{0,1,\dots, n\}\) 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 et al. [loc. cit.] 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.

### MSC:

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

60C05 | Combinatorial probability |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

60D05 | Geometric probability and stochastic geometry |

05A16 | Asymptotic enumeration |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B27 | Critical phenomena in equilibrium statistical mechanics |