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

TL;DR

This paper fully characterizes the hierarchical metastability structure of the reversible inclusion process across three time scales, confirming the conjecture and demonstrating no additional meaningful scales exist.

Contribution

It provides a complete analysis of the third time scale of metastability in the reversible inclusion process, resolving a longstanding conjecture.

Findings

Identified the third and final metastability time scale.

Proved no other significant time scales exist beyond the three.

Established sharp asymptotics for capacities using potential-theoretic and martingale methods.

Abstract

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.