Optimal corner-regularity threshold and endpoint-merging crossover

Determine whether the corner-regularity threshold used to establish the Gaussian limit for the conformal boundary statistic is optimal, and characterize how the resulting compact probability law changes when the conformal-map endpoints merge on an L-dependent scale.

Background

The paper proves the thermodynamic Gaussian and compact heat-kernel limits for a conformal boundary coordinate whose derivative has integrable inverse-square-root singularities at four marked endpoints. The proof relies on Fourier decay of order |m|^{-3/2}, finite homogeneous H^{1/2} energy, and a cutoff-removal argument for the associated discrete Toeplitz determinants.

The authors explicitly identify two unresolved issues: whether the regularity assumption used in the proof is sharp, and how the limiting compact law behaves when endpoint separations shrink with system size. Such double-scaling regimes could interpolate between the finite-energy Szegő setting treated in the paper and Fisher–Hartwig counting-statistics behavior.