Random volumes from matrices

Abstract

We propose a class of models which generate three-dimensional random volumes, where each configuration consists of triangles glued together along multiple hinges. The models have matrices as the dynamical variables and are characterized by semisimple associative algebras A. Although most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only (and all of the) three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate large N limit. We examine the analytic properties when A is a matrix ring or a group ring, and show that the models with matrix ring have a novel strong-weak duality which interchanges the roles of triangles and hinges. We also give a brief comment on the relationship of our models with the colored tensor models.

Applying the tempered Lefschetz thimble method to the Hubbard model away from half filling

“…More generally, we should keep developing the algorithm further so that it can be more easily applied to the three major problems listed in Introduction. There should also be other interesting branches of fields where the TLTM may shed new light on the theoretical understanding through a numerical analysis, such as the Chern-Simons theory [26] and matrix models that generate random volumes [27]. In order to understand the difficulty to find such an intermediate value of flow time that avoids both the sign and multimodal problems (without tempering), let us see the right panel of Fig.…”

Parallel tempering algorithm for integration over Lefschetz thimbles

“…It thus provides us with a good test of versatility to check whether correct results are obtained for such models where the thimble structure or its usefulness is not clear. As a related model, the numerical study of the triangle-hinge model [19,20,21] should also be interesting. The model is a sort of matrix model that generates 3D random volumes as a collection of triangles and hinges.…”

“…The model is a sort of matrix model that generates 3D random volumes as a collection of triangles and hinges. In order to restrict the resulting configurations to tetrahedral decompositions, one needs to introduce a special form of interaction [19], which makes the action complex-valued (M. Fukuma, S. Sugishita, and N. Umeda, manuscript in preparation). A numerical study was made for a simplified model (with no restriction to tetrahedral decompositions), and the existence of a third-order phase transition is confirmed (manuscript in preparation).…”

Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity

“…We find that the system has a critical dimension, given by six, over which it becomes unstable due to the wrong sign of the scalar kinetic term. In six dimensions, de Sitter spacetime becomes a solution to the EOM, signaling the emergence of a conformal symmetry, while the time evolution of the scale factor is power-law in dimensions below six.1 See, however,[4,5,6] for a matrix-model-like approach to three-dimensional quantum gravity. 2 When coupling many U(1)-fields, the authors in [12] found a promise of a phase transition higher than first order, which, however, is in conflict with the result in [13].3 As well, a certain minisuperspace model of GR can be derived from CTM [19].…”