Negative moments of the CREM partition function in the high temperature regime

TL;DR

This paper investigates the negative moments of the CREM partition function in the high temperature phase, showing they are comparable to the expectation of the inverse partition function raised to a power, uniformly in system size.

Contribution

It establishes a uniform bound relating negative moments of the CREM partition function to its expectation in the high temperature regime.

Findings

Negative moments are comparable to the inverse expectation.

Results hold uniformly for all system sizes.

Applicable for all <_c in the high temperature phase.

Abstract

The continuous random energy model (CREM) was introduced by Bovier and Kurkova in 2004 which can be viewed as a generalization of Derrida's generalized random energy model. Among other things, their work indicates that there exists a critical point $\beta_c$ such that the partition function exhibits a phase transition. The present work focuses on the high temperature regime where $\beta<\beta_c$. We show that for all $\beta<\beta_c$ and for all $s>0$, the negative $s$ moment of the CREM partition function is comparable with the expectation of the CREM partition function to the power of $-s$, up to constants that are independent of $N$.