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We solve the random energy model when the energies of the configurations takeonly integer values. In the thermodynamic limit, the average overlaps remainsize dependent and oscillate as the system size increases. While the extensivepart of the free energy can still be obtained by a standard replica calculationwith one step replica symmetry breaking, it is no longer possible to recoverthe overlaps in this way. A possible way to adapt the replica approach is toallow the sizes of the blocks in the Parisi matrix to fluctuate and to takecomplex values.

The discrete random energy model and one step replica symmetry breaking