Layered permutation copulas are closer to the uniform law than the diagonal
Year: 2026
Type: preprint
Abstract: Schrott conjectured that among all copulas on the unit square, the diagonal and the antidiagonal couplings are the farthest from the uniform law in the quadratic Wasserstein distance, at squared distance $1/10$. By density of permutation copulas the conjecture is implied by a statement about $n$ equally weighted grid points: for every permutation $\sigma$ of $\{0,\dots,n-1\}$, the points $\bigl((i+\frac12)/n,(\sigma(i)+\frac12)/n\bigr)$ are no farther from the uniform law than the diagonal points. We prove this, with strict inequality, for every layered permutation, that is, every permutation obtained by cutting $0,\dots,n-1$ into runs of consecutive values and reversing some of the runs, other than the identity and the full reversal. The proof re-cuts the optimal partition of the identity on the union of the moved strips, writes the gain as a weighted sum of two-cell comparisons, and settles each comparison by a dispersive-order inequality between the laws of $x_1+x_2$ and $x_1-x_2$ on a diagonal band. The same re-cut, done in two steps inside a window of three or four strips, gives the strict inequality for every permutation that splits into blocks of consecutive values of length at most eight (each block reversed or permuted arbitrarily), once $n\ge9$, and of length at most nine once $n\ge10$; for blocks of length four to nine the positivity of explicit functionals is proved with interval arithmetic. For $n\le10$ the inequality is proved for every permutation by an exact rational computation with Laguerre cells.
Author: Shmalo, Yitzchak
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225642321