Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles \& \(L\)-functions (Q6592967)

Given a matrix \(U\) in SO\((2 N)\) or Sp\((2 N)\) sampled according to the Haar measure with eigenvalues \(e^{ \pm i \theta_1}, \ldots, e^{ \pm i \theta_N}\), consider its characteristic polynomial\N\[\N\psi_U(\theta)=\prod_{j=1}^N\left(1-e^{-i\left(\theta-\theta_j\right)}\right)\left(1-e^{-i\left(\theta+\theta_j\right)}\right).\N\]\NDefine\N\[\N\mathfrak{L}_N^G(s, h)=\mathbb{E}_N\left[\left|\psi_U(0)\right|^{2 s-h}\left|\psi_U^{\prime \prime}(0)\right|^h\right].\N\]\NThe main result of the paper is that\N\[\N\lim _{N \rightarrow \infty} \frac{\mathfrak{L}_N^{S p}(s, h)}{N^{\frac{s(s+1)}{2}}+2 h}=\frac{2^{\frac{s^2}{2}} G(1+s) \sqrt{\Gamma(1+s)}}{\sqrt{G(1+2 s) \Gamma(1+2 s)}} \mathbb{E}\left[\left(\frac{1}{2} M\left(s+\frac{1}{2}, \frac{1}{2}\right)+1\right)^h\right],\N\]\Nwhere \(G\) is the Barnes \(G\)-function, \(\big\lfloor\vphantom{\frac32} {s+\frac32}\big\rfloor>h\geq0\), and for \(a \geq b>0\),\N\[\NM(a, b)=\lim_{N\to\infty}\sum_{j=1}^N \frac{1}{N^2 x_j^{(N)}}\N\]\Nfor points \(\big(x_1^{(N)},\ldots,x_N^{(N)}\big)\) sampled according to the Jacobi ensemble\N\[\N\begin{aligned} \mu_N^{(a, b)}(d \mathbf{x})= & \prod_{j=0}^{N-1} \frac{\Gamma(a+b+N+j+1)}{\Gamma(a+j+1) \Gamma(b+j+1) \Gamma(j+2)} \prod_{j=1}^N x_j^a\left(1-x_j\right)^b \\\N& \times \prod_{1 \leq j<k \leq N}\left(x_k-x_j\right)^2 d \mathbf{x} . \end{aligned}\N\]\NFurthermore, the \(\tau\)-function corresponding to \(M(a, b)\), which is defined as\N\[\N\tau_{a, b}(t):=t \frac{d}{d t} \log \mathbb{E}\left[e^{-t M(a, b)}\right],\N\]\Nsolves the \(\sigma\)-Painlevé III system\N\[\N\left(t \frac{d^2 \tau_{a, b}(t)}{d t^2}\right)^2+4\left(\frac{d \tau_{a, b}}{d t}\right)^2\left(t \frac{d \tau_{a, b}}{d t}-\tau_{a, b}(t)\right)-\left(a \frac{d \tau_{a, b}}{d t}+1\right)^2=0\N\]\Nfor all \(t>0\), with boundary conditions\N\[\N\begin{cases}\tau_{a, b}(0)=0 & \text { for } a>0, \\\N\left.\frac{d}{d t} \tau_{a, b}(t)\right|_{t=0}=-\frac{1}{4 a} & \text { for } a>1 .\end{cases}\N\]\NThe author also proves similar asymptotics for\N\[\N\mathfrak{R}_N^G(s, h)=\mathbb{E}_N\left[\left.\left|\mathcal{Z}_U(0)\right|^{2 s}\left|\frac{d}{d \theta} \log \mathcal{Z}_U(\theta)\right|_{\theta=0}\right|^h\right],\N\]\Nwhere\N\[\N\mathcal{Z}_U(\theta)=e^{\frac{i N}{2}(\theta+\pi)-i \sum_{k=1}^N \frac{\theta_k}{2}} \prod_{j=1}^N\left(1-e^{-i\left(\theta-\theta_j\right)}\right)\N\]\Nis the \(\mathcal{Z}\)-function corresponding to the eigenvalues on the upper part of the unit circle.