A note on the capacity estimate in metastability for generic configurations

TL;DR

This paper advances capacity estimates in metastability for arbitrary configurations by leveraging graph theory for exact pre-factor computation and combining geometric function theory with Thompson's principle for bounds, avoiding explicit test functions.

Contribution

It introduces a novel graph-theoretic approach for exact capacity pre-factor calculation and enhances existing methods with bounds using geometric function theory and Thompson's principle.

Findings

Exact computation of capacity pre-factor for arbitrary configurations

Upper bounds on capacity using geometric function theory and Thompson's principle

Avoidance of explicit test function constructions

Abstract

In this paper we further develop the ideas from Geometric Function Theory initially introduced in [arXiv:2206.13206], to derive capacity estimate in metastability for arbitrary configurations. The novelty of this paper is twofold. First, the graph theoretical connection enables us to exactly compute the pre-factor in the capacity. Second, we complete the method from [arXiv:2206.13206] by providing an upper bound using Geometric Function Theory together with Thompson's principle, avoiding explicit constructions of test functions.