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Entropy of random walk range

Itai BenjaminiGady KozmaAriel YadinAmir Yehudayoff — 2010

Annales de l'I.H.P. Probabilités et statistiques

We study the entropy of the set traced by an -step simple symmetric random walk on ℤ. We show that for ≥3, the entropy is of order . For =2, the entropy is of order /log2. These values are essentially governed by the size of the boundary of the trace.

Supercritical self-avoiding walks are space-filling

Hugo Duminil-CopinGady KozmaAriel Yadin — 2014

Annales de l'I.H.P. Probabilités et statistiques

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory γ between two points on the boundary of a finite subdomain of d is proportional to μ - length ( γ ) . When μ is supercritical (i.e. μ l t ; μ c where μ c is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

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