An $N^{-7/10}$ Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy
For every fixed inverse temperature $\beta>1$, we prove that the expected free energy per spin of the zero-field Gaussian Sherrington-Kirkpatrick model differs from its Parisi limit by at most $C_\beta N^{-7/10}$. This improves the exponent $2/3$ of our earlier work (arXiv:2609.09103). We also compare the bound with the finite-size correction of order $N^{-3/4}$ predicted by Franz, Parisi, and Ricci-Tersenghi, and with the strict improvement over $2/3$ that would follow from the ideas in OpenAI's recently released manuscripts.
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