Level sets percolation on chaotic graphs

TL;DR

This paper investigates the critical percolation behavior of level sets of eigenvectors on d-regular graphs, extending quantum chaos percolation concepts to graph theory and proving a critical threshold using a random Gaussian waves model.

Contribution

It introduces a random waves ensemble on d-regular trees to analyze level set percolation, establishing a critical threshold and connecting quantum chaos with graph eigenvector properties.

Findings

Identification of a critical level depending on eigenvalues and degree d

Proof of a critical threshold in the random Gaussian waves model

Numerical validation of the theoretical predictions

Abstract

One of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum-chaotic billiards and maps in the semi-classical limit display critical percolation. Here we extend these studies to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical computations show that the statistics of the largest level sets (the maximal connected components of the graph for which the eigenvector exceeds a prescribed value) depend critically on the level. The critical level is a function of the eigenvalue and the degree d. To explain the observed behavior we study a random Gaussian waves ensemble over the d-regular tree. For this model, we prove the existence of a critical threshold. Using the local tree property of d-regular graphs, and assuming the (local) applicability of the random waves model, we can compute the critical percolation level and reproduce the numerical simulations. These results support the random-waves model for random regular graphs and provides an extension to Bogomolny's percolation model for two-dimensional chaotic billiards.