On Random Allocation Models in the Thermodynamic Limit

TL;DR

This paper analyzes phase transitions and critical phenomena in the random allocation model, providing a unified framework for understanding its thermodynamic properties and filling gaps in previous theoretical results.

Contribution

It offers a comprehensive, self-contained derivation of the statistical properties and singularities of the model's thermodynamic potentials at critical points.

Findings

Uncovered new relationships between thermodynamic potentials.

Clarified the behavior of the model at critical points.

Discussed a quasi-probabilistic normalization approach.

Abstract

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