Gaussian-like fixed point and variational properties of integral discriminants

TL;DR

This paper explores integral discriminants associated with polynomial actions, revealing fixed-point and variational properties that generalize classical distributions, and introduces improved numerical methods for approximating related partition functions and moments.

Contribution

It establishes a fixed-point identity and variational principles for integral discriminants, extending Gaussian extremality to higher-degree polynomial actions and enhancing numerical approximation techniques.

Findings

Fixed-point identity relating moments to polynomial coefficients

Extension of Gaussian extremality to even-degree polynomial actions

Enhanced numerical convergence for partition function approximations

Abstract

We consider partition functions Z(g) = exp (-g(x))dx where g is a nonnegative polynomial action (a degree-2n form) vanishing only at the origin. Such integrals, known as integral discriminants, appear in statistical mechanics, quantum field theory, and the theory of exponential families. We show that the associated Boltzmann measure d$\mu$ = exp(-g(x))dx satisfies a fixed-point property identity relating in a simple manner its degree-2n moments to the coefficients of g. This generalizes familiar identities for the exponential distribution (degree-1) on the positive orthant and the Gaussian measure (degree-2). We further show that g is characterized by three variational principles, including a maximum-entropy principle under scaled moments constraints, extending the Gaussian extremality principle to arbitrary even-degree homogeneous actions. Exploiting these identities in a truncatedmoment numerical scheme (known as the Moment-SOS hierarchy), strengthens the standard semidefinite relaxations, and results in a much faster convergence, thus allowing more efficient approximations of the partition function Z(g) as well as moments of $\mu$.