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 generalization 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 →∞ scaling limit, this reduces to the Airy kernel.

中文翻译：

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β = 2 Stieltjes–Wigert 随机矩阵集合的全局和局部缩放限制

高斯酉系综的特征值概率密度函数 (PDF) 与具有对势的经典对数气体的玻尔兹曼因子有一个众所周知的类比 -日志 |X - 是的|, 受限于单体谐波势。一个概括是用-日志 |辛(π(X - 是的)/大号)|. 生成的 PDF 首次出现在统计物理学文献中，与非相交的布朗步行者有关，时间间隔相等吨 = 0，随后在 Calogero-Sutherland 类型的量子多体系统的研究中，以及在 Chern-Simons 场论中。它是基于 Stieltjes-Wigert 多项式的具有相关核的行列式点过程的示例。我们着手确定这个系综的矩，并根据一个特定的小数找到一个精确的表达式q-雅可比多项式。从他们的大ñ形式，可以计算全局密度。以前的工作已经根据 Ramanujan (q-Airy）功能。我们展示了如何在特定的大号 →∞缩放限制，这减少到 Airy 内核。