On Random Allocation Models in the Thermodynamic Limit

We discuss the phase transition and critical exponents in the random allocation model (urn model) for different statistical ensembles. We provide a unified presentation of the statistical properties of the model in the thermodynamic limit, uncover new relationships between the thermodynamic potentials and fill some lacunae in previous results on the singularities of these potentials at the critical point and behaviour in the thermodynamic limit. The presentation is intended to be self-contained, so we carefully derive all formulae step by step throughout. Additionally, we comment on a quasi-probabilistic normalisation of configuration weights which has been considered in some recent studies