Quasipolynomial-time algorithms for Gibbs point processes

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities. for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval [0,lambda]. As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all lambda< 1/(e(B+1) C-phi(<^>)) where C-phi(<^>) is a temperedness parameter and B is the stability constant of f phi 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(phi) f, where Delta(phi) f is the potential-weighted connective constant of the potential f. 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.