This paper is the final one in a series in which we investigate some m
odels of an interacting Bose gas using Varadhan's large deviation vers
ion of Laplacian asymptotics; in it we study the equilibrium thermodyn
amics of the full diagonal model of a Bose gas. We obtain a formula ex
pressing the pressure, in the thermodynamic limit, as the supremum of
a functional over the space of positive bounded measures. We analyse t
his formula for a large class of interaction kernels and show that the
re is a critical temperature below which there is Bose Einstein conden
sation.