Capturing the phase diagram of (2+1)-dimensional CDT using a balls-in-boxes model

TL;DR

This paper investigates a balls-in-boxes model to replicate the phase diagram of (2+1)-dimensional causal dynamical triangulations, revealing a connection to Hořava-Lifshitz gravity and the conditions for extended geometries.

Contribution

It demonstrates that a specific potential in the BIB model reproduces CDT's extended phase, linking it to Hořava-Lifshitz gravity through minisuperspace reduction.

Findings

The BIB model reproduces CDT's phase diagram with an extended phase.

A particular potential is necessary for droplet condensation in the BIB model.

The results connect CDT to Hořava-Lifshitz gravity via a minisuperspace approach.

Abstract

We study the phase diagram of a one-dimensional balls-in-boxes (BIB) model that has been proposed as an effective model for the spatial-volume dynamics of (2+1)-dimensional causal dynamical triangulations (CDT). The latter is a statistical model of random geometries and a candidate for a nonperturbative formulation of quantum gravity, and it is known to have an interesting phase diagram, in particular including a phase of extended geometry with classical properties. Our results corroborate a previous analysis suggesting that a particular type of potential is needed in the BIB model in order to reproduce the droplet condensation typical of the extended phase of CDT. Since such a potential can be obtained by a minisuperspace reduction of a (2+1)-dimensional gravity theory of the Ho\v{r}ava-Lifshitz type, our result strengthens the link between CDT and Ho\v{r}ava-Lifshitz gravity.