# Cluster expansions with renormalized activities and applications to colloids

## TL;DR

This paper develops a cluster expansion method with renormalized activities for a binary colloidal system, providing convergence conditions and applying it to hard spheres to rigorously analyze depletion interactions.

## Contribution

It introduces a novel cluster expansion approach with renormalized activities for multi-body effective interactions in colloids, improving convergence criteria based on surface properties.

## Key findings

- Proved convergence of the cluster expansion for the effective large-object system.
- Derived conditions involving surface area for convergence, improving previous volume-based criteria.
- Applied the method to hard spheres, rigorously analyzing depletion interactions.

## Abstract

We consider a binary system of small and large objects in the continuous space interacting via a non-negative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction) we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.