arXiv 2102.116012021-02-23math.PR

首段渗流割集的大偏差原理和最大流量的下大偏差原理

Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation

Barbara Dembin, Marie Théret

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摘要：

我们考虑标准的第一次通过渗流模型在重新标度格$\mathbb Z^d/n$的$d\geq 2$和有界域$Ω$在$\mathbb R^d$。我们用$Γ^1$和$Γ^2$表示$\partial Ω$的两个不相交的子集，分别代表源和汇，\textit{i.e.}，其中水可以进入$Ω$并从$Ω$逃逸。割集是将$Ω$中的$Γ^1$与$Γ^2$分开的边的集合，它具有由其边的容量之和给出的容量。在对$Ω$和边的容量分布作一定的假设下，我们已经知道极小割集序列$(\mathcal E_n^{min})_{n\geq 1}$的一个大数定律：序列$(\mathcal E_n^{min})_{n\geq 1}$几乎必然收敛于各向异性网络中极小割集的连续确定性问题的解的集合.本文的目的是导出割集的一个大偏差原理，并利用压缩原理导出PDMATHAMO中最大流的一个下大偏差原理。

英文摘要：

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $Ω$ and escape from $Ω$. A cutset is a set of edges that separates $Γ^1$ from $Γ^2$ in $Ω$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $Ω$ and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets $(\mathcal E_n^{min})_{n\geq 1}$: the sequence $(\mathcal E_n^{min})_{n\geq 1}$ converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in $Ω$.

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