Recent probabilistic results on covariant Schr\"odinger operators on infinite weighted graphs

TL;DR

This paper reviews recent probabilistic findings on covariant Schr"odinger operators on weighted graphs, highlighting applications like semiclassical limits and comparing these results to analogous theories on Riemannian manifolds.

Contribution

It clarifies the relationship between probabilistic results on weighted graphs and their smooth manifold counterparts, advancing understanding in both discrete and continuous settings.

Findings

Probabilistic methods applied to covariant Schr"odinger operators on graphs.

Connections established between graph results and Riemannian manifold analogues.

Insights into semiclassical limits in the context of weighted graphs.

Abstract

We review recent probabilistic results on covariant Schr\"odinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian manifolds.