倾斜圆柱体最大流量的上大偏差
Upper large deviations for maximal flows through a tilted cylinder
摘要：
考虑$\ZZ^d$中标准的第一通道渗流模型，对于$d\geq 2$，我们研究了柱体从上半部分到下半部分（分别从顶部到底部）的最大流量，柱体的基是边长与$n$成比例的超矩形，柱体的高度是$h(n)$，对于一定的高度函数$h$ .我们用$τ_n$（分别用$ϕ_n$）表示这个最大流量。我们要强调的是，圆筒可能是倾斜的。我们看看这些流量的概率，由表面的基础上的圆柱体，大于$ν(\vec{v})+\eps$为一些积极的$\eps$，其中$ν(\vec{v})$是几乎肯定的限制，重新缩放变量$τ_n$时，$n$走向无穷大。一方面，我们证明了这种概率的衰减速度的情况下，变量$τ_n$取决于尾部的边缘的能力的分布：它可以指数衰减快速与$n^{d-1}$，或与$n^{d-1} \min(n,h(n))$，或在中间政权。另一方面，我们证明了在可变$ϕ_n$的情况下，只要边的容量定律允许一个指数矩，这个概率就随着圆柱的体积指数地快速衰减;然而该结果的重要性受到以下事实的限制：$ν(\vec{v})$通常不是重新缩放的最大流量$ϕ_n$的几乎确定的极限，但是至少当圆柱体的高度$h(n)$与$n$相比可以忽略时是这种情况。
英文摘要：
We consider the standard first passage percolation model in $\ZZ^d$ for $d\geq 2$ and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to $n$ and whose height is $h(n)$ for a certain height function $h$. We denote this maximal flow by $τ_n$ (respectively $ϕ_n$). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than $ν(\vec{v})+\eps$ for some positive $\eps$, where $ν(\vec{v})$ is the almost sure limit of the rescaled variable $τ_n$ when $n$ goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable $τ_n$ depends on the tail of the distribution of the capacities of the edges: it can decays exponentially fast with $n^{d-1}$, or with $n^{d-1} \min(n,h(n))$, or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable $ϕ_n$ decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that $ν(\vec{v})$ is not in general the almost sure limit of the rescaled maximal flow $ϕ_n$, but it is the case at least when the height $h(n)$ of the cylinder is negligible compared to $n$.
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