### Cluster expansions: necessary and sufficient convergence conditions

Probability Seminar

16th March 2022, 4:00 pm – 5:00 pm

Online,

Correlation functions of Gibbs measures in general cannot be computed

explicitly, but at low density / for weak interactions they are

amenable to a power series expansion around the ideal gas (Poisson

point process). The convergence of these series has been studied at

least since the 60s, but sufficient convergence conditions keep being

developed, raising the question if there is a theoretical limit to the

improvements - an if and only if condition that cannot be surpassed.

The talk presents such an if and only if condition for repulsive

interactions. It is based on Kirkwood-Salsburg equations and

alternating sign properties and it generalizes a result by Bissacot,

Fernández and Procacci. For possibly attractive potentials, the

criterion yields new sufficient convergence conditions. Based on joint

work with Leonid Kolesnikov (arXiv:2112.13134 [math.ph]).

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