Universal Lebesgue permuton limit for GEPP permutations

Prove that for every nondegenerate Wigner matrix or iid matrix ensemble with an entry distribution fixed independently of the dimension, the permutation induced by Gaussian elimination with partial pivoting converges weakly in probability to Lebesgue measure on the unit square, under the stated pivot-comparison and tie-breaking conventions.

Background

The paper shows that the GEPP permutation is not exactly uniform for finite dimensions in the GOE, GUE, and iid Bernoulli models, as demonstrated through the leading-pivot probability. Numerical diagrams nevertheless suggest that the associated permutation measures converge globally to the same Lebesgue permuton as uniformly random permutations. The conjecture extends this observed behavior to broad classes of Wigner and iid ensembles, despite their finite-dimensional non-uniformity.

References

This suggests the following broader conjecture: for broad classes of random matrix ensembles, the permutation induced by GEPP has a universal global geometry, even when its finite-n distribution is not uniform.

— Random permutations using GEPP  (2610.10481 - Gunawan et al., 7 Oct 2026) in Conjecture 1, Section 2 (following Figure 2)

Extending it to GSE and other $\beta$-ensembles not covered by \Cref{thm:unif_perm} remains open.

— Random permutations using GEPP  (2610.10481 - Gunawan et al., 7 Oct 2026) in Section 6, “Discussion and future directions”