Fourth-Order Fusion Asymptotics for Sineβ Correlation Functions
Abstract
We compute the fourth-order correction to the full-collision asymptotics of the correlation functions of the Sineβ process. For m ≥ 2 and mβ> 3, the normalized correlation has an expansion through order ε4, 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(ε4), 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 mβ> 3 without requiring fourth moments of every analytic derivative. For general unitary ensembles with a C4 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 β, we retain a conditional quadratic transfer criterion.
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BibTeX
@misc{fang2026fourthorder,
title = {{Fourth-Order Fusion Asymptotics for $\mathrmSine_β$ Correlation Functions}},
author = {Weiyang Fang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11543},
eprint = {2609.11543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11543},
}
RIS
TY - UNPB
TI - Fourth-Order Fusion Asymptotics for Sine_β Correlation Functions
AU - Weiyang Fang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11543
UR - https://arxiv.org/abs/2609.11543
AB - We compute the fourth-order correction to the full-collision asymptotics of the correlation functions of the Sine_β process. For m ≥ 2 and mβ> 3, the normalized correlation has an expansion through order ε^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(ε^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 mβ> 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 β, we retain a conditional quadratic transfer criterion.
ER -