标题： 随机覆盖的熵和四维量子引力 显示英文标题

标题： Entropy of random coverings and 4D quantum gravity

摘要： 我们讨论了在固定欧拉示性数和基本群的给定表示下的有界几何的$n$维黎曼流形的最小测地线球覆盖计数，以及 Reidemeister 扭量。 这种计数与离散量子引力模型的连续极限分析有关。 我们建立了覆盖数随体积呈指数增长的条件，从而允许寻找相应离散化模型的连续极限。 所得的熵估计通过相应的 Reidemeister 扭量依赖于流形的基本群的表示。 我们讨论了在二维和四维情况下的非等价表示之和。 在两种情况下都得到了明确的熵函数以及相关临界指数的重要界限。 显示英文摘要

摘要： We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.