Phase diagram of the mean field model of simplicial gravity

TL;DR

This paper analyzes the phase diagram of a mean-field model of simplicial gravity, identifying phase transitions and a new condensed phase influenced by weight functions and density bounds.

Contribution

It introduces a detailed analysis of phase transitions in a mean-field simplicial gravity model with variable weights and density constraints, revealing new phases and transition behaviors.

Findings

Two phases: elongated (fluid) and crumpled.

First order transition for β > 2.

Continuous transition for 1 < β ≤ 2.

Abstract

We discuss the phase diagram of the balls in boxes model, with a varying number of boxes. The model can be regarded as a mean-field model of simplicial gravity. We analyse in detail the case of weights of the form $p(q) = q^{-\beta}$, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases~: {\em elongated} ({\em fluid}) and {\em crumpled}. For $\beta\in (2,\infty)$ the transition between these two phases is first order, while for $\beta \in (1,2]$ it is continuous. The transition becomes softer when $\beta$ approaches unity and eventually disappears at $\beta=1$. We then generalise the discussion to an arbitrary set of weights. Finally, we show that if one introduces an additional kinematic bound on the average density of balls per box then a new {\em condensed} phase appears in the phase diagram. It bears some similarity to the {\em crinkled} phase of simplicial gravity discussed recently in models of gravity interacting with matter fields.