  An interesting problem in statistical physics is the condensation of
classical particles in droplets or clusters when the pair-interaction is given
by a stable Lennard-Jones-type potential. We study two aspects of this problem.
We start by deriving a large deviations principle for the cluster size
distribution for any inverse temperature $\beta\in (0,\infty)$ and particle
density $\rho\in(0,\rho_{\mathrm{cp}})$ in the thermodynamic limit. Here
$\rho_{\mathrm{cp}}>0$ is the close packing density. While in general the rate
function is an abstract object, our second main result is the
$\Gamma$-convergence of the rate function toward an explicit limiting rate
function in the low-temperature dilute limit $\beta\to \infty$,
$\rho\downarrow0$ such that $-\beta^{-1}\log\rho\to\nu$ for some
$\nu\in(0,\infty)$. The limiting rate function and its minimisers appeared in
recent work, where the temperature and the particle density were coupled with
the particle number. In the decoupled limit considered here, we prove that just
one cluster size is dominant, depending on the parameter $\nu$. Under
additional assumptions on the potential, the $\Gamma$-convergence along curves
can be strengthened to uniform bounds, valid in a low-temperature, low-density
rectangle.
