19 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…

Sine trajectories. As the number of particles tends to infinity, every scaled particle trajectory should be uniformly approximated by a sine curve with random amplitude and phase.…

A rectangular tableau is a standard Young tableau of rectangular shape, and a tableau is realizable when its entry ordering is induced by the entries of a rank-$1$ matrix, as in th…

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.

Sine curve conjecture. The trajectories of particles are close to random sine curves with high probability.

Archimedean path conjecture. For every $t$, the random permutation $\mathbf{RSN}^n_{\lfloor tN\rfloor}$ converges to the deterministic permuton $(\mathcal{A}(0),\mathcal{A}(t))$; e…

Archimedes-limit conjecture. The random $m$-out-of-$n$ subnetwork satisfies convergence in distribution