4 problems

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let $\xi^{(n)}$ be the sum of independent rando…

Let $S$ denote the multiset of ordinates of the nontrivial zeros of the Riemann zeta function. For $a>0$, define the continuous function $H_a:\mathbb{R}_{>0}\to\mathbb{C}$ by … whe…

Let $B,C>0$, let $0<\delta<1$, and let $X=(X_{ij})$ be sampled uniformly from $\mathcal M_{n,\delta}(B,C)$. Let $B_c$ denote the critical value. Phase-transition conjecture. There…

Let $X\subseteq\mathbb{R}^m$ be compact with positive Lebesgue measure, and let $\mu$ be Lebesgue measure restricted to $X$ and rescaled to have total mass one. Let…