  This study explores the relationship between the precise asymptotics of the
level-two large deviation rate function and the behavior of metastable
stochastic systems. Initially identified for overdamped Langevin dynamics
(Ges{\`u} et al., SIAM J Math Anal 49(4), 3048-3072, 2017), this connection has
been validated across various models, including random walks in a potential
field. We extend this connection to condensing zero-range processes, a complex
interacting particle system.
  Specifically, we investigate a certain class of zero-range processes on a
fixed graph $G$ with $N > 0$ particles and interaction parameter $\alpha > 1$.
On the time scale $N^2$, this process behaves like an absorbing-type diffusion
and converges to a condensed state where all particles occupy a single vertex
of $G$ as $N$ approaches infinity. Once condensed, on the time scale
$N^{1+\alpha}$, the condensed site moves according to a Markov chain on $G$,
showing metastable behavior among condensed states. The time scales $N^2$ and
$N^{1+\alpha}$ are called the pre-metastable and metastable time scales. It is
conjectured that this behavior is encapsulated in the level-two large deviation
rate function $\mathcal{I}_N$ of the zero-range process. Specifically, it is
expected that the $\Gamma$-expansion of $\mathcal{I}_N$ can be expressed
as:$$\mathcal{I}_N = \frac{1}{N^2} \mathcal{K} + \frac{1}{N^{1+\alpha}}
\mathcal{J},$$ where $\mathcal{K}$ and $\mathcal{J}$ are the level-two large
deviation rate functions of the absorbing diffusion processes and the Markov
chain on $G$. We rigorously prove this $\Gamma$-expansion by developing a
methodology for $\Gamma$-convergence in the pre-metastable time scale and
establishing a link between the resolvent approach to metastability (Landim et
al., J Eur Math Soc, 2023. arXiv:2102.00998) and the $\Gamma$-expansion in the
metastable time scale.
