# Order of Fluctuations of the Free Energy in the SK Model at Critical Temperature

## TL;DR

This paper analyzes the fluctuations of the free energy in the SK spin glass model at critical temperature, showing the variance scales as (log N)^2 and depends on the temperature approach rate.

## Contribution

It provides an elementary proof of the fluctuation order of free energy in the SK model at and near the critical temperature, including precise variance bounds.

## Key findings

- Variance of free energy at critical temperature is O((log N)^2)
- Variance near critical temperature depends on approach rate, O((log N)^2 + N^{1-α})
- Results clarify fluctuation behavior in spin glass models at phase transition

## Abstract

We present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature. It is proved that at the critical temperature the variance of the free energy is of $O((\log N)^2).$ In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate $O(N^{-\alpha})$ for some $\alpha>0,$ then the variance is of $O((\log N)^2+N^{1-\alpha}).$