论文标题
随机图的统计模型中圆的出现
Emergence of the Circle in a Statistical Model of Random Cubic Graphs
论文作者
论文摘要
我们考虑通过随机$ 3 $定义的统计模型定义的欧几里得量子重力的正式离散化，并使用ollivier曲率（RICCI曲率的粗略模拟）进行制作。数值分析表明，模型方法的Hausdorff和光谱尺寸$ 1 $在联合经典 - 热动力学限制中，我们认为该模型的缩放限制是半径$ r $，$ s^1_r $的圆。给定轻度的运动限制，这些主张可以通过完全的数学严格证明：确切地说，可以证明，对于$ 3 $的吉尔特，至少$ 4 $的$ 3 $等级图，任何最小化配置的顺序在Gromov-Hausdorff的范围内收敛于$ s^1_r $。我们还通过分析有限尺寸效应的分析，为存在二阶相变存在提供了有力的证据。这个 - 本质上是可解决的 - 一维几何形状的玩具模型是作为随机平坦表面非扰动定义的可控范例。
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.
