Sublinearity of the travel-time variance for dependent first-passage percolation

• Autores: Jacob van den Berg, Demeter Kiss
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 40, Nº. 2, 2012, págs. 743-764
• Idioma: inglés
• DOI: 10.1214/10-AOP631
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• Resumen

  • Let E be the set of edges of the d-dimensional cubic lattice ℤd, with d ≥ 2, and let t(e), e ∈ E, be nonnegative values. The passage time from a vertex v to a vertex w is defined as infπ : v→w ∑e∈π t(e), where the infimum is over all paths π from v to w, and the sum is over all edges e of π.

    Benjamini, Kalai and Schramm [2] proved that if the t(e)’s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex v is sublinear in the distance from 0 to v. This result was extended to a large class of independent, continuously distributed t-variables by Benaïm and Rossignol [1].

    We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)’s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical “Ising landscape.”