Random permutations, permanents and optimal transport

Given a function \$c\$ on the unit square, consider a probability measure on permutations where the likelihood of \$\pi\$ is proportional to \$\prod_{i=1}^n e^{-c(i/n,\pi(i/n))}\$. Under certain assumptions on \$c\$, Soumik Pal conjectured that the normalizing constant is a permanent with asymptotic behavior
$$
Z_n\sim e^{nL}C
$$
where the constants \$L\$ and \$C\$ are related to a regularization of the optimal transport problem with cost \$c\$. In joint work with Starr, we prove the conjecture when \$c\$ is block uniform. In this talk, I will sketch what makes this problem interesting to me and describe a few ideas behind our proof.