Level sets percolation on chaotic graphs

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.

References in corpus (7)

• Random wave functions and percolation
• SLE description of the nodal lines of random wave functions
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE
• Gaussian Waves on the Regular Tree
• Geometric characterization of nodal domains: the area-to-perimeter ratio
• The Statistics of the Points Where Nodal Lines Intersect a Reference Curve
• Eigenvalue spacings for regular graphs