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 $\mathbb{Q}.$ First, for Haar-distributed unitary matrices~$U$, the permanent $\perm(U)$ follows a circularly-symmetric complex Gaussian distribution $\mathcal{CN}(0,\sigma^2)$ -- 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 $\alpha$-stable distribution with stability index $\alpha\approx 1.0$--$1.4$, well below the Gaussian value $\alpha=2$. We test Aaronson's conjecture that $|\perm(X)|^2$ is asymptotically lognormal for Gaussian~$X$: it is plausible for the complex Ginibre and GOE ensembles, but appears to fail for GUE and real Ginibre, where the $\alpha$-stable tails prevent convergence. Anti-concentration, however, holds for all Gaussian ensembles and is more robust than for Haar unitaries. 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 $f(t)=\frac{1}{n}\ln|\perm(\gamma(t))|$ that is independent of~$n$ in the large-$n$ limit, with a midpoint value \[ \perm(\gamma({\textstyle\frac12})) = (-1)^{(n-1)/2}\cdot 2e^{-n}\bigl(1+\tfrac{1}{3n}+O(n^{-2})\bigr) \] for odd~$n$ and zero for even~$n$. We also study the geodesic forom the identity to the DFT matrix.