A -convergence of level-two large deviation for metastable systems: The case of zero-range processes

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 > 1. On the time scale N², 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+, the condensed site moves according to a Markov chain on G, showing metastable behavior among condensed states. The time scales N² and N^1+ 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 IN of the zero-range process. Specifically, it is expected that the -expansion of IN can be expressed as: IN = 1N² K + 1N^{1+} J, where K and J are the level-two large deviation rate functions of the absorbing diffusion processes and the Markov chain on G. We rigorously prove this -expansion by developing a methodology for -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 -expansion in the metastable time scale.