On Kirkwood-Salsburg solutions at criticality

TL;DR

This paper analyzes the spectral properties of the Kirkwood-Salsburg operator at critical points, establishing a Laurent expansion and asymptotic behavior of correlation functions, with implications for phase transition analysis.

Contribution

It proves a Laurent expansion with a simple pole for the resolvent at the dominant eigenvalue and characterizes the asymptotic limits of correlation functions at criticality.

Findings

Laurent expansion with pole of order 1 at eigenvalue

Asymptotic limit of correlation functions as activity approaches zero

Spectral radius and convergence radius determined for positive potentials

Abstract

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.