Pal's permanent conjecture: proof for block uniform matrices

Abstract

Consider a symmetric function $\mathcal{C}(x,y)$ on $[0,1]\times [0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $\mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let $A^{(n)}=(a_{i,j}^{(n)}:i,j\in [n])$ be the matrix with entries $a_{i,j}^{(n)}=\exp (-\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\mathrm{perm}\left(A^{(n)}\right)/n!\sim \exp (n\hat{I}›[\mathcal{C}])/\sqrt{\mathcal{D}[\mathcal{C}]}$$ as $n\to \infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $\hat{I}›[\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is ${\int }_0^1{\int }_0^1(\hat{I}\pm (x)+{\hat{I}}^2(y))dxdy$ where $\hat{I}\pm (x)+{\hat{I}}^2(y)$ is chosen such that $\ddot{I}(x,y):=\exp (-\mathcal{C}(x,y)-\hat{I}\pm (x)-{\hat{I}}^2(y))$ has uniform marginals. The algebraic term $\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: ${\det }_F(I+J-T^{\ast }T)$, the Fredholm determinant, where $I$ is the identity on $L^2([0,1])$, $Jf(x)\equiv {\int }_0^{1}f(z)dz$ (for all $x$) and $Tf(x)={\int }_0^1\ddot{I}(x,y)f(y)dy$. We prove the conjecture for functions $\mathcal{C}$ that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.

11 pages