随机三次图统计模型中圆的出现
Emergence of the Circle in a Statistical Model of Random Cubic Graphs
摘要：
我们考虑一个正式的离散化定义的随机$3$ -正则图的统计模型，并利用奥利维尔曲率，一个粗糙的模拟的里奇曲率的欧几里德量子引力。数值分析表明，在经典-热力学联合极限下，模型的Hausdorff维数和谱维数都接近$1$，并且我们认为模型的标度极限是半径为$r$，$S^1_r$的圆.给定温和的运动学约束，这些要求可以证明与充分的数学严谨性：准确地说，它可以表明，$3$ -正则图的周长至少$4$，任何序列的行动最小化配置收敛的意义上的Gromov-Hausdorff到$S^1_r$。我们还提出了强有力的证据，通过有限尺寸效应的分析存在一个二阶相变。这个基本上可解的一维几何模型是指作为随机平面的非微扰定义的可控范式。
英文摘要：
We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random $3$-regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature. Numerical analysis shows that the Hausdorff and spectral dimensions of the model approach $1$ in the joint classical-thermodynamic limit and we argue that the scaling limit of the model is the circle of radius $r$, $S^1_r$. Given mild kinematic constraints, these claims can be proven with full mathematical rigour: speaking precisely, it may be shown that for $3$-regular graphs of girth at least $4$, any sequence of action minimising configurations converges in the sense of Gromov-Hausdorff to $S^1_r$. We also present strong evidence for the existence of a second-order phase transition through an analysis of finite size effects. This -- essentially solvable -- toy model of emergent one-dimensional geometry is meant as a controllable paradigm for the nonperturbative definition of random flat surfaces.
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