upt上场地渗透的概率和临界指数

Pub Date : 2022-01-28 DOI:10.4153/S0008414X22000554
Laurent M'enard
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引用次数: 0

摘要

摘要在均匀无限平面三角剖分(upt)上导出了伯努利点渗流的三个临界指数。首先,我们显式地计算根簇无穷大的概率。因此,我们证明了upt上站点渗透的非临界指数为$\beta = 1/2$。然后建立了根簇顶点数生成函数的积分公式。我们用这个公式来证明,在临界情况下,根簇至少有n个顶点的概率会像$n^{-1/7}$那样衰减。最后,我们还导出了根簇周长定律的表达式,并利用它建立了在临界时,根簇周长等于n的概率像$n^{-4/3}$那样衰减。在这三个指数中,只有最后一个指数是已知的。我们的主要工具是所谓的渗透簇的垫片分解,随机玻尔兹曼映射的一般性质,以及分析组合学。
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Percolation probability and critical exponents for site percolation on the UIPT
Abstract We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$ . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$ . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$ . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.
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