Large deviation bounds for the volume of the largest cluster in 2D critical percolation
Large deviation bounds for the volume of the largest cluster in 2D critical percolation
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二维临界渗透中最大团簇体积的大偏差界限
DOI：
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发表时间：
2014
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通讯作者：
Demeter Kiss
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文献类型：
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作者：
Demeter Kiss
设M_n表示在边长为n的三角形格子上，位渗流最大簇中的位数。本文给出了当x为1且n为大值且α 1 = 5/48且C>0时，M_n / mathbb{E} M_n > x的形式为exp（-Cx^{2/α 1}）的概率的上下界.我们的结果推广到其他二维格，并加强了以前已知的指数上限由Borgs，Chayes，Kesten和Spencer [BCKS 99]。此外，在类似于文献[BCKS 99]的一般假设下，我们得到了d > 2的上界.
Let $M_n$ denote the number of sites in the largest cluster in site percolation on the triangular lattice inside a box side length $n$. We give lower and upper bounds on the probability that $M_n / mathbb{E} M_n > x$ of the form $exp(-Cx^{2/alpha_1})$ for $x geq 1$ and large $n$ with $alpha_1 = 5/48$ and $C>0$. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions $d > 2$.