6 problems

Let $\sigma^n$ be a uniform random $n$-element sorting network, and let $\sigma^n_G(i,t)$ denote the trajectory of particle $i$, with time rescaled to $[0,1]$ and space rescaled to…

Let $k$ be a fixed finite row, and consider the local regime of random sorting networks near row $k$. The associated local swaps are obtained from the rescaled jumps of Poissonized…

Embed each permutation $\tau\in S_n$ as $\overline{\tau}=(\tau(1),\ldots,\tau(n))\in\mathbb{R}^n$. These points lie on the sphere … where … and … Embedding the edges of the sorting…

Let $\sigma^n$ be a uniform random $n$-element sorting network, and define the rescaled time-$t$ permutation measure by … Let $\mathfrak{Arch}$ be the probability measure on…

Archimedean distribution conjecture. The nonzero entries of this partial permutation matrix are distributed according to the Archimedean distribution.

Let $\omega=(s_1,\dots,s_N)$ be a uniformly random sorting network of size $n$, and fix a pattern $\gamma$ of size $k$. By the preceding theorem, there are constants $c_1,c_2>0$ de…