Phase diagram of the mean field model of simplicial gravity

Details Phase diagram of the mean field model of simplicial gravity Phase diagram of the mean field model of simplicial gravity

1998-08-04

arxiv.org/abs/gr-qc/9808011v1

Nucl.Phys. B542 (1999) 413-424

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

15 pages, 5 figures

Scientific paper

10.1016/S0550-3213(98)00842-6

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.

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