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Does the free energy variance in the Sherrington-Kirkpatrick model at criticality obey a universal scaling law that interpolates continuously between the replica-symmetric and full replica symmetry breaking regimes?

Related: Parisi Formula (Talagrand-Panchenko theorem), Guerra-Toninelli interpolation inequality, Wigner semicircle law and Tracy-Widom universality at spectral edges

The Sherrington-Kirkpatrick spin glass model exhibits a phase transition at its critical temperature, and near this transition the variance of the free energy is expected to capture deep information about the landscape of the disordered system. The open problem is to determine whether the free energy variance at and near criticality satisfies a precise scaling law that smoothly connects the high-temperature replica-symmetric phase to the low-temperature full replica symmetry breaking phase, and to identify the universal exponents governing this interpolation. This would require characterizing the variance not just at isolated parameter values but as a continuous functional of the inverse temperature across the critical window.

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