KestenMcKay law for the Markoff surface mod p
Abstract
For each prime $p$, we study the eigenvalues of a 3regular graph on roughly $p^2$ vertices constructed from the Markoff surface. We show they asymptotically follow the KestenMcKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1811.00113
 arXiv:
 arXiv:1811.00113
 Bibcode:
 2018arXiv181100113D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Group Theory;
 Mathematics  Spectral Theory;
 11D25;
 05C50;
 11F72;
 37P25
 EPrint:
 24 pages, 1 figure