Scaling limits of discrete copulas are bridged Brownian sheets

TL;DR

This paper proves that as the size of a random permutation matrix grows, the associated discrete copula converges to a bridged Brownian sheet, revealing a universal scaling limit in high-dimensional permutation structures.

Contribution

It establishes a Donsker-type central limit theorem for discrete copulas, connecting permutation matrices to continuous stochastic processes in the limit.

Findings

Convergence of scaled discrete copulas to a bridged Brownian sheet.

Provides a probabilistic limit theorem for large permutation matrices.

Bridges discrete combinatorial structures with continuous stochastic processes.

Abstract

For large $n$, take a random $n \times n$ permutation matrix and its associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, let $y_n(\frac{a}{n},\frac{b}{n}) = \frac{1}{n} ( X_{a,b} - \frac{ab}{n} )$; define $y_n: [0,1]^2 \to R$ by interpolating quadratically on squares of side $\frac{1}{n}$. We prove a Donsker type central limit theorem: $\sqrt{n} y_n$ approaches a bridged Brownian sheet on the unit square.