  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.
