Sharp ultraviolet growth beyond the Hilbert–Schmidt threshold

Determine the sharp ultraviolet growth of the modulated partition function for logarithmic and Riesz interactions at and beyond the Hilbert–Schmidt threshold, including matching bounds, leading constants, and whether a renormalized limiting object governs the corresponding modulated Gibbs ensembles.

Background

For Riesz interactions with exponent s\ge d/2, the paper proves that the modulated partition function diverges, but only obtains polylogarithmic growth at s=d/2 and stretched-exponential-in-logarithm lower bounds for d/2<s<d. The authors state that these rates are probably nonoptimal under a local nondegeneracy condition on the background density.

The unresolved problem is to prove the conjectured larger rates, identify their leading constants, and determine whether an appropriate renormalization produces a meaningful limiting object beyond the Hilbert–Schmidt regime.

References

The larger rates predicted under the local nondegeneracy hypothesis in \cref{rem:expectedSharpUltravioletGrowth} remain open, as do the leading constants and the question whether a renormalized limiting object governs the corresponding modulated Gibbs ensembles.

— Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime  (2609.02743 - Delgadino et al., 2 Sep 2026) in Section 5, subsection “Consequences and open problems”; Remark 2.14, “Expected sharp ultraviolet growth”

It remains natural to ask for sharper asymptotics, including the first correction term and whether rare near-collision configurations, with pair separations below the typical interparticle scale $N{-1/d}$, produce additional local contributions not encoded by the limiting Gaussian second chaos and its Carleman--Fredholm determinant.

— Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime  (2609.02743 - Delgadino et al., 2 Sep 2026) in Section 5, subsection “Consequences and open problems”

The sharpness of this interpolation is not addressed here.

— Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime  (2609.02743 - Delgadino et al., 2 Sep 2026) in Section 5, subsection “Consequences and open problems”; Corollary 2.8 and Remark 2.9