Inspired by techniques and results from computer science and the geometry of polynomials, we turn to a classical question in statistical physics: for what range of parameters do classical particles in the continuum interacting via a pair potential remain in a gaseous state? Or in other words, for which parameters is the infinite volume pressure analytic? The classical approach to this question is to study convergence of the cluster expansion, but this can be limited by the presence of singularities of the pressure away from the positive real axis in the complex plane. In the case of a repulsive pair potential, and using ideas from Weitz and Peters-Regts, we prove analyticity of the pressure in a complex neighborhood that improves the classical bound on absence of phase transition by a factor e^2. Joint work with Marcus Michelen.