  Mayer cluster expansion is an important tool in statistical physics to
evaluate grand canonical partition functions. It has recently been applied to
the Nekrasov instanton partition function of $\mathcal{N}=2$ 4d gauge theories.
The associated canonical model involves coupled integrations that take the form
of a generalized matrix model. It can be studied with the standard techniques
of matrix models, in particular collective field theory and loop equations. In
the first part of these notes, we explain how the results of collective field
theory can be derived from the cluster expansion. The equalities between free
energies at first orders is explained by the discrete Laplace transform
relating canonical and grand canonical models. In a second part, we study the
canonical loop equations and associate them to similar relations on the grand
canonical side. It leads to relate the multi-point densities, fundamental
objects of the matrix model, to the generating functions of multi-rooted
clusters. Finally, a method is proposed to derive loop equations directly on
the grand canonical model.
