The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited

Adrien Schertzer

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arXiv · 2606.21360

The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited

Adrien Schertzer

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Abstract

We study the fluctuations of the free energy of the Sherrington-Kirkpatrick model at the critical inverse temperature $\beta=1$. We prove that \[ \operatorname{Var}(F_N(1))\leq \frac14\log N+C, \] with $C$ independent of $N$. This gives a logarithmic upper bound, in agreement with the order predicted in the physics literature. We also prove that the critical variance diverges: more precisely, \[ \operatorname{Var}(F_N(1)) \geq \frac12\log\log\log N - C . \]

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Adrien Schertzer. 2026-06-19. The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited. https://arxiv.org/abs/2606.21360

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