arXiv 0907.54992009-07math.PR

首段渗流中$\bolds{\mathbb{R}^d}$区域最大流的上大偏差

Upper large deviations for the maximal flow through a domain of $\bolds{\mathbb{R}^d}$ in first passage percolation

Raphaël Cerf, Marie Théret

备注：: Published in at http://dx.doi.org/10.1214/10-AAP732 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Applied Probability 2011, Vol. 21, No. 6, 2075-2108
DOI：: 10.1214/10-AAP732

摘要：

我们考虑标准的第一次通过渗流模型在重新标度图$\mathbb {Z}^d/n$的$d\geq2$和一个域$Ω$的边界$Γ$在$\mathbb {R}^d$。设$Γ^1$和$Γ^2$是$Γ$的两个不相交的开子集，表示$Γ$中某些水可以通过其进入和逸出$Ω$的部分。我们研究了流$ϕ_n$通过相应的离散集$Γ^1_n$和$Γ^2_n$之间的$Ω$的离散版本$Ω_n$的渐近行为。我们证明了在区域的正则性和边的容量律的条件下，$ϕ_n/n^{d-1}$的上大偏差在某一常数以上是体积阶的，即随着$n^d$指数快速衰减.这篇文章是一个更大的项目的一部分，在这个项目中，作者证明了这个常数是a.s. $ϕ_n/n^{d-1}$的限度。

英文摘要：

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of $ϕ_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of $ϕ_n/n^{d-1}$.

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摘要：

英文摘要：

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