On the accuracy of the normal approximation for the free energy in the Random Energy Model

Meiners, Raphael; Reichenbachs, Anselm

doi:10.1214/ecp.v18-2377

Cited by 1 publication

(1 citation statement)

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“…in the range where the contour C 1 crosses the real axis). If not one can deform the contour to pass through the saddle point but one should not forget the contribution of the poles at the integer values of r. This is what happens in particular in the high temperature phase [24,42].…”

Section: Some Remarks On the Replica Calculationmentioning

confidence: 99%

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

Derrida¹,

Mottishaw²

2015

J. Stat. Mech.

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We present a systematict 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.

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Section: Some Remarks On the Replica Calculationmentioning

confidence: 99%