Integrals over Grassmannians and Random permutations

TL;DR

This paper explores complex integrals over Grassmannians related to the distribution of sample canonical correlations in Gaussian populations, connecting them to hypergeometric functions, Painlevé equations, and combinatorial structures.

Contribution

It introduces a novel connection between integrals over Grassmannians, hypergeometric functions, Painlevé equations, and combinatorial expansions involving random words.

Findings

Integral representations relate to hypergeometric functions.

Connections established with Painlevé equations.

Distribution characterized via combinatorial expansions.

Abstract

In testing the independence of two Gaussian populations, one computes the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The "Laplace transform" of this distribution is not only an integral over the Grassmannian of p-dimensional planes in complex n-space, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlev\'e-like equations. They also have expansions, related to random words of length l formed with an alphabet of p letters. Given that each letter appears in the word, the maximal length of the disjoint union of p increasing subsequences of the word clearly equals l. But the maximal length of the disjoint union of p-1 increasing subsequences leads to a non-trivial distribution. It is precisely this probability which appears in the expansion above.