Forbidden Gaps at a One-Sided Fractional Edge: Two-Cut Equilibrium and Exact Pressure

This preprint studies forbidden-gap formation at a one-sided fractional spectral edge in a convex unitary random-matrix ensemble. Starting from an explicit critical equilibrium density with a cube-root vanishing at the left endpoint, we analyze the constrained equilibrium problem produced by forbidding eigenvalues in an interval adjacent to that edge.

We prove that the constraint generates a two-cut equilibrium measure with a newly occupied component on the opposite side of the critical endpoint, determine the associated endpoint scales, and derive an exact moving-hard-edge pressure identity. In particular, we obtain the remote-edge displacement law and the leading constrained free-energy asymptotic

Igap(a)−I∗=Cgapa8/3+o(a8/3),I_{\mathrm{gap}}(a)-I_* = C_{\mathrm{gap}}a^{8/3}+o(a^{8/3}),

with the coefficient CgapC_{\mathrm{gap}} determined explicitly through the unique solution of a period-matching problem.

The final part of the paper connects this equilibrium geometry with the twelve-ray fractional-edge determinantal process developed in the companion preprint “A Twelve-Ray Fractional-Edge Process in a Convex Unitary Ensemble” (Zenodo record 22136099, DOI: 10.5281/zenodo.22136099). The equilibrium period condition is shown to coincide with the scalar band-matching condition arising in the associated integrable-kernel/Riemann–Hilbert formulation. The undeformed large-hole asymptotic for the limiting determinantal process is identified as a subsequent problem and is not claimed here.