Critical and subcritical fusion asymptotics for Sine${}_{\beta }$ correlation functions

Abstract

We determine the first correction to the leading Vandermonde fusion law for the correlation functions of the Sine$\ast \beta$ process in the critical and and subcritical regimes $m\beta \le 1$. When $m\beta <1$, the normalized correction is of order $|\epsilon {|}^{1+m\beta }$, with a strictly negative coefficient given by an explicit gamma-function ratio times an absolutely convergent arithmetic-geometric mean deficit integral. At $m\beta =1$, it is $-\frac{\sum \ast i<j(a_i-a_j{)}^2}{8m^2(2m+1)}{\epsilon }^2\log (1/|\epsilon |)+O({\epsilon }^2)$. For two merging points, the subcritical coefficient reduces to a gamma-function expression involving $\sec (\pi \beta )-1$, and the critical logarithmic coefficient is $-1/160$. The argument starts from a geometric interpolation of circular-Jacobi weights. Selecting one particle in the interpolation derivative increases the fused charge by $\beta$ and leaves a strict positive-power moment margin in the remaining stochastic-zeta expectation. This yields an absolutely convergent one-particle identity and uniform control at the collision scale. The results resolve the critical and subcritical conjecture in the author's earlier preprint and complement its supercritical second-order expansion.

9 pages. Companion paper to arXiv:2608.23742