  One of the main contributions of this paper is to illustrate how large
deviation theory can be used to determine the equilibrium distribution of a
basic droplet model that underlies a number of important models in material
science and statistical mechanics. The model is simply defined. $K$
distinguishable particles are placed at random onto the $N$ sites of a lattice,
where the ratio $K/N$, the average number of particles per site, equals a
constant $c \in (1,\infty)$. We focus on configurations for which each site is
occupied by at least one particle. The main result is the large deviation
principle (LDP), in the limit where $K \rightarrow \infty$ and $N \rightarrow
\infty$ with $K/N = c$, for a sequence of random, number-density measures,
which are the empirical measures of dependent random variables that count the
droplet sizes. The rate function in the LDP is the relative entropy $R(\theta |
\rho^*)$, where $\theta$ is a possible asymptotic configuration of the
number-density measures and $\rho^*$ is a Poisson distribution restricted to
the set of positive integers. This LDP reveals that $\rho^*$ is the equilibrium
distribution of the number-density measures, which in turn implies that
$\rho^*$ is the equilibrium distribution of the random variables that count the
droplet sizes. We derive the LDP via a local large deviation estimate of the
probability that the number-density measures equal $\theta$ for any probability
measure $\theta$ in the range of these random measures.
