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

Free energy of disordered urn models in the canonical ensemble