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, . 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 . 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.

我们考虑由随机 3-正则图的统计模型定义的欧几里得量子引力的形式离散化，并利用奥利维尔曲率，即 Ricci 曲率的粗略模拟。数值分析表明，模型的 Hausdorff 和谱维数在联合经典热力学极限中接近 1，我们认为模型的尺度极限是半径为r的圆，. 给定温和的运动学约束，这些主张可以用完全的数学严谨性来证明：准确地说，可以证明对于至少为 4 的 3-正则图，任何最小化配置的动作序列在 Gromov-Hausdorff 的意义上收敛到. 我们还通过对有限尺寸效应的分析，提供了存在二阶相变的有力证据。这个——本质上是可解的——涌现一维几何的玩具模型意味着作为随机平面的非微扰定义的可控范式。