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.