Percolation Hamiltonians

TL;DR

This paper surveys recent progress on the spectral properties of percolation Hamiltonians, highlighting interdisciplinary mathematical approaches to understanding their asymptotic behavior.

Contribution

It compiles and explains key results on spectral asymptotics of random operators on percolation subgraphs, integrating graph theory, group theory, probability, and operator theory.

Findings

Summarizes advances in spectral asymptotics of percolation Hamiltonians

Connects results across multiple mathematical disciplines

Provides background and context for ongoing research

Abstract

There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators.