AI 中文总结
本文计算了 Sine$_\beta$ 过程相关函数全碰撞渐近的四阶修正，通过 Ward 恒等式和期望-Taylor 引理证明展开系数，并推广到一般酉系综。
AI 中文摘要
我们计算了 Sine$_\beta$ 过程相关函数的全碰撞渐近的四阶修正。对于 $m \ge 2$ 且 $m\beta > 3$，归一化相关函数具有直到 $\epsilon^4$ 阶的展开，其显式有理系数仅通过中心化轮廓的四次幂和与二次幂和的平方来依赖该轮廓。余项为 $o(\epsilon^4)$，在碰撞轮廓上局部一致成立。我们通过有限维 Ward 恒等式计算所需的 Hua-Pickrell 环境的四阶逆矩，并利用特征多项式导数界证明其收敛性。一个四阶期望-Taylor 引理处理了 $m\beta > 3$ 的完整范围，而不需要对每个解析导数都要求四阶矩。对于具有 $C^4$ 约束势和正则体点的一般酉系综，我们利用复核普适性和差商证明了有限粒子融合系数直到四阶的收敛性以及联合二阶极限。该酉结果不施加融合环境假设。对于任意 $\beta$，我们保留了一个条件二次转移准则。
英文摘要
We compute the fourth-order correction to the full-collision asymptotics of the correlation functions of the $\mathrm{Sine}_β$ process. For $m \ge 2$ 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.
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