2010/05/24 by Yehonatan Elon, Elon, Yehonatan, Uzy Smilansky +1
Mathematics · Physics and Astronomy · #05C80 #60G15 #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #msc:05C80 #msc:60G15 #nlin.CD
paper · pdf · doi:10.48550/arxiv.1005.4322
openalex publication_date 2010/05/24 · arxiv created 2010/07/25 · arxiv updated 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.