Pal's permanent conjecture: proof for block uniform matrices
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 $A^{(n)} = \big(a^{(n)}_{i,j}\, :\, i,j \in [n]\big)$ be the matrix with entries $a^{(n)}_{i,j}\, =\, \exp(-\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\operatorname{perm}\big(A^{(n)}\big)/n!\sim \exp\big(n \Lambda[\mathcal{C}]\big)/ \sqrt{\mathcal{D}[\mathcal{C}]}$$ as $n \to \infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $\Lambda[\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is $\int_{0}^1 \int_0^{1} (\alpha(x)+\beta(y))\, dx\, dy$ where $\alpha(x)+\beta(y)$ is chosen such that $\rho(x,y) := \exp(-\mathcal{C}(x,y)-\alpha(x)-\beta(y))$ has uniform marginals. The algebraic term $\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: $\operatorname{det}_F(I+J-T^*T)$, 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 $Tf(x) = \int_0^1 \rho(x,y) f(y)\, dy$. 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.