Low-temperature zero-field SK free-energy fluctuations

Determine the fluctuation order and limiting distribution of the zero-field Sherrington–Kirkpatrick free energy \(\ln Z_N^{SK}(\beta,0)\) for every \(\beta>1\).

Background

The paper’s comparison theorem shows that, when the variance of the zero-field SK free energy diverges, the SKFI free energy in the spin-glass regime has the same fluctuations as the zero-field SK model. However, the zero-field SK fluctuation theory below the critical temperature is not known, and even the fluctuation scale remains unresolved.

The paper notes that the best available rigorous result is only an upper bound of order N/ln⁡NN/\ln N for the variance, while the divergence of the variance is expected but has not been proved.

References

For \beta>1, even the order of the fluctuations is a major open problem: the best known upper bound is \var\big(\ln Z_N{SK}(\beta,0)\big) = O(N/\ln N), while the divergence of the variance --- the hypothesis \nu_N\to\infty of Proposition~\ref{prop:SG} --- is universally expected but unproven.

— Fluctuations of the free energy of the Sherrington-Kirkpatrick model with ferromagnetic interaction  (2608.25362 - Dey et al., 26 Aug 2026) in Section 1.3, “SK model without external field”