4 problems

Let $X_1,\ldots,X_n$ be independent and identically distributed random variables with an analytic density on $(0,\infty)$, and let $X_{(k;n)}$ denote the $k$-th order statistic. Ar…

Let $n_0(N,\epsilon)$ be the smallest integer $n\geq1$ for which all eigenvalues of $U_N^n$ lie in $[e^{-i\pi\epsilon},e^{i\pi\epsilon}]$. Let $X_1,\dots,X_N$ be independent unifor…

Let $\tau_{N,\operatorname{strong}}$ be the first strong return time for the powers of an $N\times N$ random unitary matrix, and let $\widetilde{\tau}_{N,\epsilon}$ be the analogou…

Boundary-regularity conjecture. For every $r\ge2$, condition (vi) can be weakened to