14 problems

Let $\omega_n$ be an $n$-element uniform sorting network, let ${\mathbb S}_n$ denote the permutahedron, and let $d_\infty$ be the distance from a sorting network to a great circle…

Archimedes configurations. For all $t\in(0,1)$, the scaled configuration converges in distribution to the Archimedes measure in the vague topology for random Borel measures on…

For $n\geq 2$, let $\operatorname{SYT}(\delta_n)$ be the set of standard Young tableaux of staircase shape $\delta_n$, let $\operatorname{SN}_n$ be the set of sorting networks on…

For $n\ge 2$, let $\delta_n$ be the staircase Young diagram, let $\operatorname{SYT}(\delta_n)$ be its standard Young tableaux, and let $\operatorname{SN}_n$ be the set of sorting…

Let $(X_{i,j})_{i,j\ge 1}$ be an array of independent, identically distributed exponential random variables of rate $1$. For each $n\ge 2$, let $\bm{V}_n$ denote the vector of last…

AHRV frozen-region conjecture. As $n\to\infty$, the boundary of the limit shape of $F_t$ is

Adjacency distribution conjecture. The random sorting-network statistic satisfies $Y_n(\alpha)/\mathbb{E}[Y_n(1)]\to c\alpha$ in probability for some constant $c$, and the random…

Local trajectory reconstruction conjecture. For every $\epsilon>0$, there exists $C>0$ such that

Exponential concentration conjecture. There exist constants $c_1,c_2>0$ such that, for every $n$,

Local-limit conjecture. There exists a rotationally symmetric, translation-invariant point process $Pi$ on $\mathbb{R}^2$ such that, for every $(x,y)$ in the open unit disk,

Maximum fluctuation conjecture. For every $\epsilon>0$,

Fluctuation-variance conjecture. The sequence

Angel–Holroyd–Romik–Virág Archimedean-limit conjecture. For every $t\in[0,1]$,

Archimedean process convergence conjecture. $\mathbf{RSN}^n$ converges in probability, as a permutation process, to the Archimedean process.