Random permutations using GEPP

Abstract

Gaussian elimination with partial pivoting (GEPP) remains the most widely used solver for dense linear systems $A \mathbf x = \mathbf b$ for $A \in \mathbb C^{n\times n}$. We study the permutation $π= π(A)$ that arises in the GEPP factorization $PA = LU$, encoded by the permutation matrix factor $P = P_π$. When the input matrix is random, so is $π$. For random scalar butterfly matrices of size $2^n$ (a recursively defined family originally introduced to eliminate the need for pivoting altogether), we give the exact GEPP factorization and fully classify the induced permutation as an element of a $2$-Sylow subgroup of $S_{2^n}$ contained in the separable permutations. Moreover, the uniform-angle model induces the uniform distribution on this subgroup. For the GOE, GUE, and iid Bernoulli models, the induced permutation is never exactly uniform for $n \ge 2$. We give the precise rate of departure from uniformity at the leading pivot for the GOE and GUE, and give evidence that this non-uniformity vanishes asymptotically in the permuton sense. In contrast, for banded random matrices of sublinear bandwidth, including the tridiagonal $β$-Hermite ensembles, the induced permutation converges to the diagonal permuton. We further show that the resulting pivot probabilities are sensitive to implementation choices: standard LAPACK routines compare complex pivot candidates using the $\ell^1$ rather than $\ell^2$ norm, changing the GUE(2) pivot probability from $1/\sqrt3$ to $2/3$. Together these results establish a new connection between random matrix theory and permutation combinatorics through numerical linear algebra.