2 problems

Let Assumption $\mathrm{(function)}$ hold, and set $q=q(N)=e^{-\gamma(N)/N}$, where $\gamma(N)\to+\infty$ and $\gamma(N)\ll N$. Let $\lambda^r_j(N)$ denote the parts of the random…

For each $n$, let $\mathcal P_n^{GUE}(\theta,\phi)$ denote the level-crossing density for a pencil of independent GUE matrices, expressed in spherical coordinates $(\theta,\phi)$ o…