Principal minors Pfaffian half-tree theorem

A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize half-trees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian half-tree theorem, we obtain a refined version of the matrix-tree theorem in the case of skew-symmetric matrices, as well as a line-bundle version of this result.