On the accuracy of the normal approximation for the free energy in the REM

TL;DR

This paper investigates the accuracy of the normal approximation for free energy fluctuations in the random energy model, revealing its limitations at high temperatures and specific scalings.

Contribution

It provides a detailed analysis of the regimes where the normal approximation is valid and identifies the decay rate of moderate deviations.

Findings

Normal approximation holds only in a narrow scaling range at high temperatures.

Probabilities of moderate deviations decay faster than exponentially for higher order scalings.

The results clarify the limitations of CLT-based approximations in the REM.

Abstract

In the present paper we consider the fluctuations of the free energy in the random energy model (REM) on a moderate deviation scale. We find that for high temperatures the normal approximation holds only in a narrow range of scalings away from the CLT. For scalings of higher order, probabilities of moderate deviations decay faster than exponentially.