正弦β相关函数的二阶融合渐近分析
Second-order Fusion Asymptotics for Sine\b{eta} Correlation Functions
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中文总结 AI 辅助
该研究针对Sine_β过程相关函数，在mβ>1区域计算出二阶融合渐近的非平凡修正项，给出归一化二阶系数表达式，证明结合多种数学工具，副产品得到HP分布下的矩结果。
中文摘要 AI 辅助
近期研究给出了Sine_β过程相关函数的全β、全阶随机ζ表示，并确定了当多个变量合并时它们的首项范德蒙德渐近行为。我们在mβ>1的整个区域内计算首个非平凡修正项。若a₁,…,aₘ为互不相同的实数，且V(a)=∑₁≤i<j≤m(a_i−a_j)²，则当ε→0时，ρ^(m)_β(εa₁,…,εaₘ)=C^(m)_β |ε|^(β·C(m,2))·∏_(i<j)|a_i−a_j|^β·[1−β²V(a)/(8(mβ−1)(2m+1))·ε²+o(ε²)]。对于m=2，归一化二阶系数对所有β>1/2均为−β²/[40(2β−1)]。证明结合了有限N旋转Ward恒等式、精确Hua–Pickrell迹矩、随机ζ整函数及其导数的紧矩界，以及定量多元期望泰勒引理。作为副产品，我们计算得到E_{HP_{β,mβ/2}}∑_x x⁻²=mβ/[4(mβ−1)(2m+1)]。mβ=1处的极点标志着当前二阶矩论证的边界，暗示下一个融合修正项的形式存在转变。
英文摘要
Recent work gives an all-$β$, all-order stochastic-zeta representation of the correlation functions of the $\Sine_β$ process and determines their leading Vandermonde asymptotics when several variables merge. We compute the first nontrivial correction throughout the regime $mβ>1$. If $a_1,\ldots,a_m$ are distinct real numbers and \[ V(a)=\sum_{1\le i<j\le m}(a_i-a_j)^2, \] then, as $\varepsilon\to0$, \[ ρ^{(m)}_β(\varepsilon a_1,\ldots,\varepsilon a_m) =C^{(m)}_β|\varepsilon|^{β\binom m2}\prod_{i<j}|a_i-a_j|^β\left[1-\frac{β^2 V(a)}{8(mβ-1)(2m+1)}\varepsilon^2+o(\varepsilon^2)\right]. \] For $m=2$ the normalized second-order coefficient is $-β^2/[40(2β-1)]$ for every $β>1/2$. The proof combines a finite-$N$ rotational Ward identity, exact Hua--Pickrell trace moments, compact moment bounds for the stochastic-zeta entire function and its derivatives, and a quantitative multivariate expectation--Taylor lemma. As a by-product we evaluate \[ \E_{\HP_{β,mβ/2}}\sum_x x^{-2}=\frac{mβ}{4(mβ-1)(2m+1)}. \] The pole at $mβ=1$ marks the boundary of the present second-moment argument and suggests a transition in the form of the next fusion correction.
