随机三角剖分上渗流问题的Boltzmann方法
A Boltzmann approach to percolation on random triangulations
摘要：
本文利用生成函数方法研究了玻尔兹曼三角剖分上的渗流模型。更确切地说，我们认为玻尔兹曼模型上的一组有限的平面三角形，连同渗透配置（无论是网站渗透或债券渗透）在这个三角形。通过根据边界长度和边界上的顶点/边的数量来枚举具有边界的三角剖分，我们能够识别原始簇的几何形状的相变。例如，我们证明了渗流界面长度为n的概率随n呈指数衰减，但在渗流参数p的一个特定值pc处，其衰减为多项式（n -10/3阶）.此外，如果p < p c，则原始聚类具有大小n的概率呈指数衰减，如果p $\ge$ p\_c，则呈多项式衰减。位渗流的临界渗流值为pc = 1/2，键渗流的临界渗流值为pc = 2 $\sqrt$ 3- 1 11。这些值与Angel为位点渗透确定的无限三角形临界渗透阈值以及Angel \& Curien为键渗透确定的临界渗透阈值一致，并且我们给出了这些渗透阈值的独立推导。最后，我们重新讨论了随机Boltzmann映射的临界性条件，并指出在p c处，条件为大小为n的渗流簇应收敛于Le Gall & Miermont提出的参数为7 6的稳定映射.这使我们能够推导出一些新的临界指数。
英文摘要：
We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length n decays exponentially with n except at a particular value p\_c of the percolation parameter p for which the decay is polynomial (of order n --10/3). Moreover, the probability that the origin cluster has size n decays exponentially if p < p c and polynomially if p $\ge$ p\_c. The critical percolation value is p\_c = 1/2 for site percolation, and p c = 2 $\sqrt$ 3--1 11 for bond percolation. These values coincide with critical percolation thresholds for infinite triangula-tions identified by Angel for site-percolation, and by Angel \& Curien for bond-percolation, and we give an independent derivation of these percolation thresholds. Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at p\_c , the percolation clusters conditioned to have size n should converge toward the stable map of parameter 7 6 introduced by Le Gall \& Miermont. This enables us to derive heuristically some new critical exponents.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
