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March 2020 The maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$
Barbara Dembin
Ann. Probab. 48(2): 622-645 (March 2020). DOI: 10.1214/19-AOP1367

Abstract

We consider the standard first passage percolation model on $\mathbb{Z}^{d}$ with a distribution $G$ on $\mathbb{R}^{+}$ that admits an exponential moment. We study the maximal flow between a compact convex subset $A$ of $\mathbb{R}^{d}$ and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut $A$ from infinity. We prove that the rescaled maximal flow between $nA$ and infinity $\phi (nA)/n^{d-1}$ almost surely converges toward a deterministic constant depending on $A$. This constant corresponds to the capacity of the boundary $\partial A$ of $A$ and is the integral of a deterministic function over $\partial A$. This result was shown in dimension $2$ and conjectured for higher dimensions by Garet in (Annals of Applied Probability 19 (2009) 641–660).

Citation

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Barbara Dembin. "The maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$." Ann. Probab. 48 (2) 622 - 645, March 2020. https://doi.org/10.1214/19-AOP1367

Information

Received: 1 July 2018; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199856
MathSciNet: MR4089489
Digital Object Identifier: 10.1214/19-AOP1367

Subjects:
Primary: 60K35
Secondary: 82B43

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • March 2020
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