  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.
