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

TL;DR

This paper develops an exact method to compute finite size corrections in the low temperature phase of the random energy model, connecting replica symmetry breaking, complex replica numbers, and saddle point analysis.

Contribution

It introduces a systematic, exact approach to finite size corrections in the REM, linking replica symmetry breaking assumptions with complex integral representations.

Findings

Explicit finite size corrections for overlap functions are derived.

Replica symmetry breaking can be understood through complex replica numbers.

The approach explains the convergence of certain divergent series.

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.