论文标题
磁盘随机三角剖分的模型：相变
Ising model on random triangulations of the disk: phase transition
论文作者
论文摘要
在[Arxiv：1806.06668]中，我们研究了磁盘的玻尔兹曼随机三角剖分，耦合到其面部的ISING模型，在其临界温度下具有Dobrushin边界条件。在本文中，我们通过将先前的结果扩展到任意温度来研究该模型的相变：我们在所有温度下计算模型的分区函数，并得出与无限外围限制相关的几个关键指数。我们表明，该模型在任何温度下都具有局部限制，其特性巨大取决于温度。在高温下，局部极限让人联想到以亚临界渗透装饰的均匀的无限半平面三角剖分（UIHPT）。在低温下，局部极限会由于托布鲁什（Dobrushin）边界条件施加的两个旋转簇之间的主要iSing接口的能源成本而形成有限宽度的瓶颈。可以通过具有不错的几何含义的新订单参数来概括此变化。除了相变外，我们还将[ARXIV：1806.06668]中使用的两步渐进状态的局部限制构建到更自然的对角线渐近方制度。我们在该方案中获得了与主要ISIS界面长度相关的缩放限制，这与量子表面连续理论（又称\ liouville量子重力）的预测相吻合。
In [arXiv:1806.06668], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [arXiv:1806.06668] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a.\ Liouville quantum gravity).
