Critical and subcritical fusion asymptotics for Sineβ correlation functions
Abstract
We determine the first correction to the leading Vandermonde fusion law for the correlation functions of the Sine*β process in the critical and and subcritical regimes mβ≤1. When mβ<1, the normalized correction is of order |ε|1+mβ, with a strictly negative coefficient given by an explicit gamma-function ratio times an absolutely convergent arithmetic-geometric mean deficit integral. At mβ=1, it is −(∑*i<j(ai−aj)2)/(8m2(2m+1))ε2log(1/|ε|)+O(ε2). For two merging points, the subcritical coefficient reduces to a gamma-function expression involving sec(πβ)−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 β 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.
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BibTeX
@misc{fang2026critical,
title = {{Critical and subcritical fusion asymptotics for Sine$_β$ correlation functions}},
author = {Weiyang Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07239},
eprint = {2609.07239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07239},
}
RIS
TY - UNPB
TI - Critical and subcritical fusion asymptotics for Sine_β correlation functions
AU - Weiyang Fang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07239
UR - https://arxiv.org/abs/2609.07239
AB - We determine the first correction to the leading Vandermonde fusion law for the correlation functions of the Sine*β process in the critical and and subcritical regimes mβ≤1. When mβ<1, the normalized correction is of order |ε|^1+mβ, with a strictly negative coefficient given by an explicit gamma-function ratio times an absolutely convergent arithmetic-geometric mean deficit integral. At mβ=1, it is −(∑*i<j(a_i−a_j)^2)/(8m^2(2m+1))ε^2log(1/|ε|)+O(ε^2). For two merging points, the subcritical coefficient reduces to a gamma-function expression involving sec(πβ)−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 β 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.
ER -