标题： 几何空间的自组装来自随机图 显示英文标题

标题： Self-Assembly of Geometric Space from Random Graphs

摘要： 我们提出了一种欧几里得量子引力模型，在该模型中，随机图动态自组装成离散流形结构。具体来说，我们考虑了一个由欧几里得爱因斯坦-希布特作用离散化驱动的统计模型；与基于单纯复形和Regge微积分的传统方法不同，我们的离散化基于奥利维耶曲率，这是流形Ricci曲率的一种粗略类比，适用于一般的图。由于奥利维耶曲率的定义基于最优传输理论，通常难以计算，但我们为一类相关图中的奥利维耶曲率给出了一个全新的精确表达式，仅涉及边上的短循环数量。这一结果应独立地引起网络理论家的兴趣。最小化作用的构型证明是至多缺陷的三次复形；有迹象表明，这种缺陷在宏观极限下被动态抑制。进一步研究一个无缺陷的模型显示，某些经典构型具有几何解释，并且离散地逼近了欧几里得爱因斯坦-希布特作用的真空解。在几何构型为理论稳定真空的构型空间中工作，我们得到了存在连续相变的直接数值证据；这使得该模型成为欧几里得爱因斯坦引力的UV完备化。值得注意的是，这个相变意味着出现的几何空间熵满足面积律。理论的某些真空可以解释为婴儿宇宙；我们发现这些构型在模型的平均场近似中表现为稳定真空，但只要作用精确，它们就会被动态排除，表明几何空间的动态稳定性。该模型旨在作为后续研究涌现时间机制的框架。 显示英文摘要

摘要： We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.