Permanents of matrix ensembles: computation, distribution, and geometry

Abstract: We report on a computational and experimental study of permanents. On the computational side, we use the GPU to greaatly accelerate the computation of permanents over $\mathbb{C},$ $\mathbb{R},$ $\mathbb{F}_p$ and $\mathbb{Q}.$ In particular, we use this to compute the permanents of DFT and Schur matrices far beyond the ranges hitherto known. On the experimental side, we present two new observations. First, for Haar-distributed unitary matrices~$U$, the permanent $\perm(U)$ follows a circularly-symmetric complex Gaussian distribution $\mathcal{CN}(0,σ^2)$ -- we confirm this via a number of tests for $n$ up to~23 with $50{,}000$ samples. The DFT matrix permanent is an extreme outlier for every prime $n\ge 7$. In contrast, for Haar-random \emph{orthogonal} matrices~$O$, the permanent $\perm(O)$ is approximately real Gaussian but with positive excess kurtosis that decays as~$O(1/n)$, indicating slower convergence. For matrices with Gaussian entries (GUE, GOE, Ginibre), the permanent follows an $α$-stable distribution with stability index $α\approx 1.0$--$1.4$, well below the Gaussian value $α=2$. Secondly, we study the permanent along geodesics on the unitary group. For the geodesic from the identity to the $n$-cycle permutation matrix, we find a universal scaling function $f(t)=\frac{1}{n}\ln|\perm(γ(t))|$ that is independent of~$n$ in the large-$n$ limit, with a midpoint value [ \perm(γ({\textstyle\frac12})) = (-1)^{(n-1)/2}\cdot 2e^{-n}\bigl(1+\tfrac{1}{3n}+O(n^{-2})\bigr) ] for odd~$n$ and zero for even~$n$. For the geodesic to the DFT matrix, the permanent recovers $10$--$40$ times above its valley minimum when $n$ is prime, but not when $n$ is composite -- a geodesic fingerprint of primality.