论文标题
自旋簇在随机三角剖分中的几何特性与ISING模型相结合
Geometric properties of spin clusters in random triangulations coupled with an Ising Model
论文作者
论文摘要
我们在随机三角剖分中研究了典型的自旋簇的几何形状，并在有限和无限的体积设置中，与其顶点上的iSing构型的能量成比例。已知该模型在明确的临界温度下进行组合相变，其分区函数的渐近行为与均匀的地图不同。这项工作的目的是给出这种相变的几何证据。 在无限的体积设置中，称为无限的平面三角剖分，我们表现出无限自旋簇存在的相变：对于临界和超临界温度，根旋转簇几乎肯定是有限的，而它是无限的，对于亚临界温度而言，概率为正。值得注意的是，我们能够为此概率获得明确的参数表达式，该表达式允许证明渗透率临界指数为$β= 1/4 $。 我们还得出了在有限和无限体积设置中的尾部分布和根旋转簇的尾部分布的临界指数。最后，我们建立了被视为looptree的根旋转群集的界面的缩放限制。特别是在整个超临界温度方案中，我们证明了关键指数和循环限制与关键的伯努利站点渗透相同。 我们的证明混合了组合和概率论证。起点是垫片分解，它充分利用了我们模型的空间马尔可夫特性。这种分解使我们能够将根旋转簇描述为有限体积设置中的玻尔兹曼平面图。然后，我们结合了通过分析组合和玻尔兹曼地图的通用特征获得的精确组合结果，以确定我们的结果。
We investigate the geometry of a typical spin cluster in random triangulations sampled with a probability proportional to the energy of an Ising configuration on their vertices, both in the finite and infinite volume settings. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. The purpose of this work is to give geometric evidence of this phase transition. In the infinite volume setting, called the Infinite Ising Planar Triangulation, we exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. Remarkably, we are able to obtain an explicit parametric expression for this probability, which allows to prove that the percolation critical exponent is $β=1/4$. We also derive critical exponents for the tail distribution of the perimeter and of the volume of the root spin cluster, both in the finite and infinite volume settings. Finally, we establish the scaling limit of the interface of the root spin cluster seen as a looptree. In particular in the whole supercritical temperature regime, we prove that the critical exponents and the looptree limit are the same as for critical Bernoulli site percolation. Our proofs mix combinatorial and probabilistic arguments. The starting point is the gasket decomposition, which makes full use of the spatial Markov property of our model. This decomposition enables us to characterize the root spin cluster as a Boltzmann planar map in the finite volume setting. We then combine precise combinatorial results obtained through analytic combinatorics and universal features of Boltzmann maps to establish our results.
