Quasipolynomial-time algorithms for Gibbs point processes

Abstract

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighborhood of the interval $[0,\lambda ]$. As a corollary, for all finite-range stable potentials we obtain a quasipolynomial-time determinsitic algorithm for all $\lambda </(e^{B+1}{\hat{C}}_{\varphi })$ where ${\hat{C}}_{\varphi }$ is a temperedness parameter and $B$ is the stability constant of $\varphi$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $\lambda <e/{\Delta }_{\varphi }$, where ${\Delta }_{\varphi }$ is the potential-weighted connective constant of the potential $\varphi$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

Results extended to include stable potentials