Simulating four-dimensional simplicial gravity using degenerate triangulations

Abstract

We extend a model of four-dimensional simplicial quantum gravity to include degenerate triangulations in addition to combinatorial triangulations traditionally used. Relaxing the constraint that every 4-simplex is uniquely defined by a set of five distinct vertexes, we allow triangulations containing multiply connected simplexes and distinct simplexes defined by the same set of vertexes. We demonstrate numerically that including degenerated triangulations substantially reduces the finite-size effects in the model. In particular, we provide a strong numerical evidence for an exponential bound on the entropic growth of the ensemble of degenerate triangulations, and show that a discontinuous crumpling transition is already observed on triangulations of volume N 4 ≈ 4000. Discretized models of four-dimensional Euclidean quantum gravity, known as simplicial gravity or dynamical triangulations, have received ample attention in recent years, the hope being that in a suitable scaling limit they might provide a sensible non-perturbative definition of quantum gravity. In the simplicial gravity approach the integration over metrics is replaced by a sum over an ensemble of triangulations constructed by all possible gluings of equilateral 4-simplexes into closed (piece-wise linear) simplicial manifolds (see e.g. Ref. [1, 2]). The regularized Euclidean EinsteinHilbert action is particularly simple; it can be taken to depend on only two coupling constants, µ and κ, related to the cosmological and the inverse Newton's constants. The coupling constants are conjugate to the volume -the number of 4-simplexes -and the number of triangles in a given triangulation respectively. The regularized grand-canonical partition function thus becomes:The sum is over all distinct triangulations T ∈ T , N i is the number of i-simplexes in a triangulation T and C T denotes its symmetry factor -the number of equivalent labeling of the vertexes.

Euclidian 4d quantum gravity with a non-trivial measure term

“…The set of triangulations of S 4 we use are the so-called combinatorial triangulations, where every 4-simplex is uniquely defined by a set of 5 distinct vertices and by demanding that two adjacent 4-simplices share precisely one face (a three-dimensional subsimplex). This is in contrast to the degenerate triangulations, defined in [45], and used in the recent study of the crinkled phase [39]. It is believed that the models defined by combinatorial triangulations and degenerate triangulations belong to the same universality class, and using a different class of triangulations than used in [39] will give us a check of the robustness of the results obtained in [39] as well as in this study.…”

Lattice quantum gravity and asymptotic safety

“…Most early simulations of EDT used a set of triangulations that satisfies the combinatorial manifold constraints, so that each distinct 4-simplex has a unique set of 4+1 vertex labels. The combinatorial manifold constraints can be relaxed to include a larger set of degenerate triangulations in which distinct 4-simplices may share the same 4+1 (distinct) vertex labels [38]. The reason we use degenerate triangulations is that it leads to a factor of ∼10 reduction in finite-size effects compared to combinatorial triangulations [38].…”

“…The combinatorial manifold constraints can be relaxed to include a larger set of degenerate triangulations in which distinct 4-simplices may share the same 4+1 (distinct) vertex labels [38]. The reason we use degenerate triangulations is that it leads to a factor of ∼10 reduction in finite-size effects compared to combinatorial triangulations [38]. There appears to be no essential difference in the phase diagram between degenerate and combinatorial triangulations in fourdimensions [18], suggesting that if a second-order transition could be identified, the two sets of triangulations would be in the same universality class.…”

Exploring the Phase Diagram for Lattice Quantum Gravity

“…The set of combinatorial triangulations was used in most early simulations of EDT. However, the constraint of combinatorial uniqueness can be relaxed to include the larger set of degenerate triangulations in which the neighbours of a given simplex are no longer unique [15]. It has been shown numerically that simulations using degenerate triangulations lead to a factor of ∼ 10 reduction in finite size effects compared to combinatorial triangulations [15].…”

“…However, the constraint of combinatorial uniqueness can be relaxed to include the larger set of degenerate triangulations in which the neighbours of a given simplex are no longer unique [15]. It has been shown numerically that simulations using degenerate triangulations lead to a factor of ∼ 10 reduction in finite size effects compared to combinatorial triangulations [15]. We have made various checks of our code against the literature using combinatorial triangulations [16], as well as for degenerate triangulations [11], and good agreement was found.…”