  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.
