Percolating level sets of the adjacency eigenvectors of d-regular graphs

Smilansky, Uzy and Elon, Yehonatan 2010. Percolating level sets of the adjacency eigenvectors of d-regular graphs. Journal of Physics A: Mathematical and Theoretical 43 (45) 10.1088/1751-8113/43/45/455209

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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 behaviour 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 numerical simulations. These results support the random-waves model for random regular graphs, suggested in [1] and provide an extension to Bogomolny's percolation model [2] for two-dimensional chaotic billiards.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	IOP Publishing
ISSN:	1751-8121
Last Modified:	19 Mar 2016 22:28
URI:	https://orca.cardiff.ac.uk/id/eprint/17720

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