论文标题
UIPT上现场渗透的渗透概率和关键指数
Percolation probability and critical exponents for site percolation on the UIPT
论文作者
论文摘要
我们在均匀的无限平面三角剖分（UIPT）上得出了三个关键指数的伯诺利位点渗透。首先，我们明确计算根簇是无限的概率。结果，我们表明，在UIPT上的站点渗透的非临界指数为$β= 1/2 $。然后，我们为根集群中顶点数量的生成函数建立一个积分公式。我们使用此公式来证明，在关键时期，根群至少具有$ n $ VERTICES腐烂的可能性，例如$ n^{ - 1/7} $。最后，我们还得出了根群外围的定律的表达式，并使用它来确定根群的周长等于$ n $衰减的可能性，例如$ n^{ - 4/3} $。在这三个指数中，只有最后一个是以前知道的。我们的主要工具是渗透簇的所谓垫片分解，随机玻尔兹曼图的通用特性以及分析组合。
We derive three critical exponents for Bernoulli site percolation on the 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 $β= 1/2$. 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, as well as analytic combinatorics.
