Fourth-Order Fusion Asymptotics for ${\mathrm{Sine}}_{\beta }$ Correlation Functions

Abstract

We compute the fourth-order correction to the full-collision asymptotics of the correlation functions of the ${\mathrm{Sine}}_{\beta }$ process. For $m\ge 2$ and $m\beta >3$, the normalized correlation has an expansion through order ${\epsilon }^4$, with an explicit rational coefficient depending on the centered profile only through its fourth power sum and the square of its second power sum. The remainder is $o({\epsilon }^4)$, locally uniformly in the collision profile. We evaluate the required fourth inverse moments of the Hua-Pickrell environment by finite-dimensional Ward identities and prove their convergence using characteristic-polynomial derivative bounds. A fourth-order expectation-Taylor lemma handles the full range $m\beta >3$ without requiring fourth moments of every analytic derivative. For general unitary ensembles with a $C^4$ confining potential and a regular bulk point, we prove convergence of the finite-particle fusion coefficients through fourth order and a joint second-order limit, using complex kernel universality and divided differences. This unitary result imposes no fused-environment hypotheses. For arbitrary $\beta$, we retain a conditional quadratic transfer criterion.

16 pages