论文标题
$β= 2 $ stieltjes的全球和本地缩放限制 - Wigert随机矩阵合奏
Global and local scaling limits for the $β= 2$ Stieltjes--Wigert random matrix ensemble
论文作者
论文摘要
高斯单位合奏的特征值概率密度函数（PDF）具有与带有配对势$ - \ log |的经典日志气体的Boltzmann因子的众所周知的类比| x -y | $，受一体谐波潜力限制。概括是将对电位替换为$ - \ log | \ sinh（π（x-y）/l）| $。由此产生的PDF首先出现在统计物理学文献中，与非互相关的布朗步行者有关，在时间$ t = 0 $，然后在研究Calogero-Sutherland类型的量子系统中，以及Chern-Simons Field理论。这是基于stieltjes的确定点过程的一个示例，该过程具有相关性核的一个示例 - Wigert多项式。我们解决了确定该合奏时刻的问题，并根据特定的小$ q $ -jacobi多项式找到一个精确的表达。从它们的大$ n $表单中，可以计算全球密度。先前的工作已经根据Ramanujan（$ Q $ -Airy）功能评估了相关内核的边缘缩放限制。我们展示了在特定的$ l \ to \ infty $缩放限制中如何减少到通风的内核。
The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well known analogy with the Boltzmann factor for a classical log-gas with pair potential $- \log | x - y|$, confined by a one-body harmonic potential. A generalisation is to replace the pair potential by $- \log |\sinh (π(x-y)/L) |$. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time $t=0$, and subsequently in the study of quantum many body systems of the Calogero-Sutherland type, and also in Chern-Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes--Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little $q$-Jacobi polynomial. From their large $N$ form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how in a particular $L \to \infty$ scaling limit, this reduces to the Airy kernel.
