Hierarchical structure of metastability in the reversible inclusion process: third time scale and complete characterization

In this article, we study the hierarchical structure of metastability in the reversible inclusion process. We fully characterize the third time scale of metastability subject to any underlying geometry of the system and prove that this is the last time scale. We also demonstrate that there are no other meaningful time scales except the three identified ones. This work completes the verification of the conjecture made in [7] which was partially resolved on the first time scale in [7] and on the second time scale in [25]. Main tools are potential-theoretic approach and martingale approach to metastability; we thoroughly investigate the highly-complicated energy landscape of the system to construct suitable test objects to provide sharp asymptotics on capacities.