Phases of Tree-decorated Dynamical Triangulations in 3D

Abstract

This work revisits the Euclidean Dynamical Triangulation (DT) approach to non-perturbative quantum gravity in three dimensions. Inspired by a recent combinatorial study by T. Budd and L. Lionni of a subclass of 3-sphere triangulations constructed from trees, called the \emph{triple-tree} class, we present a Monte Carlo investigation of DT decorated with a pair of spanning trees, one spanning the vertices and the other the tetrahedra of the triangulation. The complement of the pair of trees in the triangulation can be viewed as a bipartite graph, called the \emph{middle graph} of the triangulation. In the triple-tree class, the middle graph is restricted to be a tree, and numerical simulations have displayed a qualitatively different phase structure compared to standard DT. Relaxing this restriction, the middle graph comes with two natural invariants, namely the number of connected components and loops. Introducing corresponding coupling constants in the action, allows one to interpolate between the triple-tree class and unrestricted tree-decorated DT. Simulations of this extended model confirm the existence of a new phase, referred to as the \emph{triple-tree phase}, besides the familiar crumpled and branched polymer phases of DT. A statistical analysis of the phase transitions is presented, showing hints that the branched polymer to triple-tree phase transition is continuous.

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