Symmetric matrices, Catalan paths, and correlations
Combinatorics
2015-11-16 v1 Statistics Theory
Statistics Theory
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.
Cite
@article{arxiv.1511.04125,
title = {Symmetric matrices, Catalan paths, and correlations},
author = {Bernd Sturmfels and Emmanuel Tsukerman and Lauren Williams},
journal= {arXiv preprint arXiv:1511.04125},
year = {2015}
}