Symmetric matrices, Catalan paths, and correlations

TL;DR

This paper proves a conjecture linking symmetric matrices to Catalan paths, providing explicit formulas for matrix entries and applying these results to a statistical correlation problem.

Contribution

It confirms the Kenyon-Pemantle conjecture, establishing a new combinatorial parametrization of symmetric matrices using Catalan paths.

Findings

Proved the Kenyon-Pemantle conjecture for symmetric matrices.

Established a bijection from the cube to the elliptope for correlation matrices.

Connected combinatorial structures to matrix parametrizations and statistical applications.

Abstract

Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.