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.
Singularities of solutions to quadratic vector equations on complex upper half-plane
设$ S $是作用在有界函数上的保正对称线性算子.在复上半平面$ \mathbb{H} $中具有参数$ z $的非线性方程$ -\frac{1}{m}=z+Sm $具有唯一解$ m $，其值在$ \mathbb{H} $中。我们表明，$ z $依赖于这个解决方案可以表示为一个家庭的概率措施$ v $ $ \mathbb{R} $的Stieltjes变换。在适当的条件下，证明了$ S $上的$ v $除了有100多个次数不超过3的代数奇点外，还有一个实解析密度.我们的动机来自大型随机矩阵。解决方案$ m $确定两个突出的矩阵系综的特征值的密度;（i）具有中心独立条目的矩阵，其方差由$ S $给出;以及（ii）具有平移不变相关结构的相关条目的矩阵。我们的分析表明，极限本征值密度只有平方根奇点或一个立方根尖点，没有其他奇点发生。
