arXiv:1005.4322v2 [math-ph] 25 Jul 2010 Percolating level sets of the adjacency eigenvectors of d-regular graphs Yehonatan Elon 1 and Uzy Smilansky 1,2 1 Department of Physics of Complex Systems, The Weizmann Institute of Science, 76100 Rehovot, Israel 2 Cardiff School of Mathematics and WIMCS, Cardiff University, Senghennydd road, Cardiff CF24 4AG, UK. 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, suggested in [1] and provides an extension to Bogomolny’s percolation model [2] for two-dimensional chaotic billiards. 1. Introduction The statistics of the adjacency spectrum of random d-regular graphs (in the limit of large vertex number) displays the generic attributes associated with the spectra of quantum Hamiltonians (in the limit  → 0) whose classical dynamics is chaotic. This observation