On Kirkwood-Salsburg solutions at criticality

In this work we study the Kirkwood-Salsburg equations of equilibrium classical continuous systems. We prove a Laurent expansion for the resolvent, at an eigenvalue of largest modulus of the Kirkwood-Salsburg operator, which is shown to have a pole of order 1. Then we prove that all correlation functions have an asymptotic limit as the activity parameter tends to a smallest zero of the partition function. As corollary, we show that any smallest zero of the partition function is simple. The main consequence is that in case of positive or hardcore potentials we find the spectral radius of the Kirkwood-Salsburg operator and the convergence radius of the solutions.