论文标题
溪流的大偏差原理和第一次通道中的最大流量
Large deviation principle for the streams and the maximal flow in first passage percolation
论文作者
论文摘要
我们考虑在$ d \ geq 2 $的重新定位晶格$ \ mathbb {z} ^d $中的标准第一段通过渗透模型和$ \ mathbb r ^d $中的有界域$ω$。我们用$γ^1 $和$γ^2 $两个不相交的子集为$ \ partialω$，分别代表源和水槽，即水可以以$ω$输入并从$ω$中逃脱。最大流是一个向量度量$ \oferrightArrowμ_n^{max} $，它描述了如何通过$γ^1 $输入的最大流体量并以$ω$传播。根据$ω$和$ g $的某些假设，我们已经知道$ \oferrightarrowμ_n^{max} $的大数字法律。序列$（\oferrightarrowμ_n^{max}）_ {n \ geq 1} $几乎可以肯定地收敛到各向异性网络中最大流的连续确定性问题的解决方案集。我们的目的是在这里得出一个较大的偏差原理，并根据收缩原理推断出最大流量的上部大偏差的速率函数的存在。
We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ 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 source and the sink, i.e., where the water can enter in $Ω$ and escape from $Ω$. A maximal stream is a vector measure $\overrightarrowμ_n^{max}$ that describes how the maximal amount of fluid can enter through $Γ^1$ and spreads in $Ω$. Under some assumptions on $Ω$ and $G$, we already know a law of large number for $\overrightarrowμ_n^{max}$. The sequence $(\overrightarrowμ_n^{max})_{n\geq 1} $ converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in $Ω$.
