Finite size corrections in the random energy model and the replica approach

Abstract

We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally, our approach allows one to see why some apparently diverging series or integrals are harmless.

arXiv comment23 pages, 1 figure, revised version 11 December 2014

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• Equilibrium Fluctuations in Mean-field Disordered Models
• On the genealogy of branching random walks and of directed polymers
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• Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models
• Finite size corrections to the Parisi overlap function in the GREM
• The discrete random energy model and one step replica symmetry breaking