A Γ-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 α > 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 N1+α, the condensed site moves according to a Markov chain on G, showing metastable behavior among condensed states. The time scales N² and N1+α 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 Γ-expansion of IN can be expressed as:IN = 1/N² K + 1N1+α 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 Γ-expansion by developing a methodology for Γ-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 Γ-expansion in the metastable time scale.