Phase diagram of the mean field model of simplicial gravity

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: elongated (fluid) and crumpled. For beta is an element of (2,infinity) the transition between these two phases is first-order, while for beta is an element of (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 condensed phase appears in the phase diagram. It bears some similarity to the crinkled phase of simplicial gravity discussed recently in models of gravity interacting with matter fields. (C) 1999 Elsevier Science B.V.