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Permanents of matrix ensembles: computation, distribution, and geometry

Igor Rivin

TL;DR

This work comprehensively studies permanents across computation, distribution, and geometry. It introduces a GPU-accelerated pipeline combining CRT-based exact computation with Gray-code Ryser evaluation to compute $perm(S_n)$ up to $n=43$, extending prior benchmarks. Empirically, $perm(U)$ for Haar-uniform $U$ follows a circularly symmetric complex Gaussian law with variance $ obreak{n!/n^n}$, while the DFT matrix is a pronounced outlier for prime $n$, and Gaussian ensembles yield heavy-tailed, $ ext{α}$-stable permanents with $ ext{α} ext{ in }[1.0,1.4]$. The study of permanents along geodesics in $U(n)$ uncovers a universal scaling function $f(t)$ for the cycle geodesic and a primality fingerprint for the $I o F_n$ geodesic, offering new insights with implications for boson sampling and the classical simulation frontier.

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.

Permanents of matrix ensembles: computation, distribution, and geometry

TL;DR

This work comprehensively studies permanents across computation, distribution, and geometry. It introduces a GPU-accelerated pipeline combining CRT-based exact computation with Gray-code Ryser evaluation to compute up to , extending prior benchmarks. Empirically, for Haar-uniform follows a circularly symmetric complex Gaussian law with variance , while the DFT matrix is a pronounced outlier for prime , and Gaussian ensembles yield heavy-tailed, -stable permanents with . The study of permanents along geodesics in uncovers a universal scaling function for the cycle geodesic and a primality fingerprint for the geodesic, offering new insights with implications for boson sampling and the classical simulation frontier.

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 and 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~, the permanent follows a circularly-symmetric complex Gaussian distribution -- we confirm this via a number of tests for up to~23 with samples. The DFT matrix permanent is an extreme outlier for every prime . In contrast, for Haar-random \emph{orthogonal} matrices~, the permanent is approximately real Gaussian but with positive excess kurtosis that decays as~, indicating slower convergence. For matrices with Gaussian entries (GUE, GOE, Ginibre), the permanent follows an -stable distribution with stability index --, well below the Gaussian value . Secondly, we study the permanent along geodesics on the unitary group. For the geodesic from the identity to the -cycle permutation matrix, we find a universal scaling function that is independent of~ in the large- limit, with a midpoint value for odd~ and zero for even~. For the geodesic to the DFT matrix, the permanent recovers -- times above its valley minimum when is prime, but not when is composite -- a geodesic fingerprint of primality.
Paper Structure (24 sections, 1 theorem, 10 equations, 8 figures, 3 tables)

This paper contains 24 sections, 1 theorem, 10 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Computational methods
  3. CRT-based exact computation
  4. Gray code block parallelization
  5. GPU acceleration
  6. Distribution of $|\mathop{\mathrm{perm}}\nolimits(U)|$ for Haar unitaries
  7. Experimental setup
  8. Complex Gaussian distribution
  9. DFT outlier for prime $n$
  10. Permanents along geodesics on $U(n)$
  11. Geodesic to the $n$-cycle
  12. Geodesic to the DFT matrix
  13. Complex trajectories
  14. Orthogonal matrices: a real counterpart
  15. Approximate normality with excess kurtosis
  16. ...and 9 more sections

Key Result

Proposition 11

If $H$ is an $n\times n$ Hermitian matrix, then $\mathop{\mathrm{perm}}\nolimits(H)\in\mathbb{R}$.

Figures (8)

  • Figure 1: Q--Q plots of $|\mathop{\mathrm{perm}}\nolimits(U)|$ against the Rayleigh distribution for Haar-random unitary matrices, $n=7,11,13,17,19,23$ ($50{,}000$ samples each). The red star marks the DFT matrix, which lies beyond the 100th percentile for every $n$ shown. Weibull shape parameters $c$ and KS $p$-values are shown in each panel title.
  • Figure 2: Complex Gaussian tests for $\mathop{\mathrm{perm}}\nolimits(U)$ with Haar-random $U\in U(n)$, $n=7,11,13,17,19,23$ ($50{,}000$ samples each). Top: scatter plots of $(\mathrm{Re},\mathrm{Im})$ with $1\sigma$--$3\sigma$ circles. Middle: Q--Q plots of Mahalanobis $d^2$ against $\chi^2_2$. Bottom: phase histograms against the uniform density.
  • Figure 3: Scaled permanent $\frac{1}{n}\ln|\mathop{\mathrm{perm}}\nolimits(\gamma(t))|$ along geodesics on $U(n)$. Left: geodesic $I\to C_n$ showing universal collapse; the dashed line marks $-1$. Right: geodesic $I\to F_n$ (DFT) for various primes.
  • Figure 4: Complex trajectory of $\mathop{\mathrm{perm}}\nolimits(\gamma(t))$ along geodesics on $U(n)$ for $n=7,11,13,17,19,23$. Top: DFT geodesic $I\to F_n$, colored by $t$; green circle marks $t=0$, red square marks $t=1$, orange diamond marks $t=1/2$. Bottom: cycle geodesic $I\to C_n$ (log scale); the permanent is real-valued, with blue/red indicating sign.
  • Figure 5: Normality test for $\mathop{\mathrm{perm}}\nolimits(O)$ with Haar-random $O\in O(n)$ ($50{,}000$ samples each). Top: histograms with fitted normal density. Bottom: Q--Q plots. Excess kurtosis is visible in the heavy tails for small $n$ but diminishes with increasing $n$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 2
  • Remark 6: Phase cancellation
  • Remark 7: Real-valuedness
  • Remark 10: Cycle geodesic on $O(n)$
  • Proposition 11
  • proof
  • Remark 13: Unitarity is essential for Gaussianity