Entropy of random coverings and 4D quantum gravity

Abstract

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.

54 pages, latex, no figures

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Cited by in corpus (10)

• Discrete approaches to quantum gravity in four dimensions
• On locally constructible spheres and balls
• Singular Vertices and the Triangulation Space of the D-sphere
• (2+1)-Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
• More on the exponential bound of four dimensional simplicial quantum gravity
• Topology in 4D simplicial quantum gravity
• Recent progress in the theory of random surfaces and simplicial quantum gravity
• On the entropy bound of three dimensional simplicial gravity
• Five Dimensional Dynamical Triangulations
• The Geometry of the Elongated Phase in 4-D Simplicial Quantum Gravity