Petter Brändén: Praise be to the Lorentz signature: Hodge-Riemann relations for the q-state Potts model

Abstract

Recall that a real symmetric matrix has Lorentz signature if it is nonsingular and has exactly one positive eigenvalue. We introduce a family of polynomials with remarkable properties, in terms of Lorentzian signature. Lorentzian polynomials generalize determinants, hyperbolic polynomials, and Minkowski volume polynomials.

We show that Lorentzian polynomials are intimately connected to independence (matroid theory) as well as negative dependence in statistical physics. Indeed we characterize matroids in terms of Lorentzian polynomials. We prove that the q-state Potts model partition function is Lorentzian when 0<q ≤1. Consequences are long sought after negative dependence properties for the q-state Potts model, as well as a proof of Mason's strongest conjecture from 1972 on independent sets in a matroid.

This is joint work with June Huh, IAS, Princeton.