Free energy of disordered urn models in the canonical ensemble

TL;DR

This paper calculates the free energy of disordered urn models using the law of large numbers, showing the replica symmetric solution is exact and revealing a mathematical similarity to the Random Field Ising Model.

Contribution

It demonstrates that the replica symmetric solution for the disordered urn model is exact by comparing it with the law of large numbers and the replica method.

Findings

Replica symmetric solution is exact for the disordered urn model.

The saddle point equations from the law of large numbers and replica method coincide.

Mathematical similarity between urn models and the Random Field Ising Model is established.

Abstract

We calculate the free energy of the disordered urn model using the law of large numbers. It is revealed that the saddle point equation obtained by the usage of the law of large numbers is the same as that obtained by the replica method. Hence, we conclude that the replica symmetric solution is adequate for the disordered urn model. Furthermore, we point out the mathematical similarity of free energies between the urn models and the Random Field Ising Model; this similarity gives an evidence that the replica symmetric solution of the urn models is exact.