Non-Abelian correlation inequalities and stable determinantal polynomials

TL;DR

This paper explores correlations in certain spin models at zero coupling, showing their asymptotic behavior relates to Kirchhoff polynomials, and proves that inverse half-integer powers of specific stable polynomials satisfy generalized correlation inequalities.

Contribution

It establishes a general theorem linking inverse half-integer powers of determinantal stable polynomials to generalized correlation inequalities, extending known results.

Findings

Correlations become inverse powers of Kirchhoff polynomials in large power limits.

Inverse half-integer powers of stable polynomials satisfy GKS and Ginibre inequalities.

Asymptotic properties relate ferromagnetic behavior to log-ultramodularity.

Abstract

We consider the correlations of invariant observables for the $O(N)$ and $\mathbb{C}\mathbb{P}^{N-1}$ models at zero coupling, namely, with respect to the natural group-invariant measure. In the limit where one takes a large power of the integrand, we show that these correlations become inverse powers of the Kirchhoff polynomial. The latter therefore provide a simplified toy model for the investigation of inequalities between products of correlations. Properties such as ferromagnetic behavior for spin model correlations correspond, in this asymptotic limit, to log-ultramodularity which is a consequence of the Rayleigh property of the Kirchhoff polynomial. In addition to the above rigorous asymptotics, the main result of this article is a general theorem which shows that inverse half-integer powers of certain determinantal stable polynomials, such as the Kirchhoff polynomials, satisfy generalizations of the GKS 2 inequalities and the Ginibre inequalities. We conclude with some open problems, e.g., the question of whether the last statement holds for powers which are not half-integers. This leads to a Hirota-bilinear analogue of the complete monotonicity property recently investigated by Scott and Sokal.