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((logN)2): In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate O(N-α) for some α > 0; then the variance is of O((logN)2 + N1-α).
Bibliographical noteFunding Information:
Both authors thank Sourav Chatterjee for explaining Talagrand’s upper bound for the variance of the free energy at the critical temperature and Erik Bates for the illuminating discussion and careful reading. They also thank the anonymous referee for providing many useful suggestions regarding the presentation of the paper. The first author’s research is partially supported by NSF grants DMS-17-52184.
© 2019 ALEA, Lat. Am. J. Probab. Math. Stat.
- Disorder chaos
- Sherrington-Kirkpatrick model
- Spin glass