6 problems

Level-spacing sum-rule conjecture.

B$^3$G sum rule. For every such pair $(n,k)$, the sum vanishes identically:

Let $\Sigma\in\mathcal M_{p,1}(\mathbb T)$ have infinite support, and let $(\boldsymbol\alpha_k)_{k\geq0}\in(\mathbb B_p)^{\mathbb N_0}$ be its matrix Verblunsky coefficients. Let…

Let $\mu\in\mathcal M_1(\mathbb T)$ have Verblunsky coefficients $(\alpha_k)_{k\geq 0}\in\mathbb D^{\mathbb N}$. Let $\theta_{\tt g}$, $N^-$, $N^+$, $\lambda_i^+$, $\lambda_i^-$, a…

Let $\widehat{J\beta E}(N,\kappa_1N,\kappa_2N)$ denote the Jacobi beta-ensemble model with parameters $\kappa_1N$ and $\kappa_2N$, let $\nu$ be a probability measure on $[0,1]$, le…

Let $\nu$ be a probability measure on $(0,\infty)$, let $MP$ denote the Marchenko–Pastur measure, let ${\mathcal K}(MP\!\ |\!\ \nu)$ denote the reversed Kullback–Leibler divergence…