Computer Science Department
School of Computer Science, Carnegie Mellon University
Level Spacings for SL(2,p)
John D. Lafferty, Daniel N. Rockmore*
To appear in Emerging Applications of Number Theory,,
The IMA Volumes in Mathematics and its Applications,
Eds.: A. Friedman, W. Miller, Jr., Springer Verlag, 1997.
Keywords: Random matrices, Cayley graphs, expander graphs,
spacing distribution, Gaussian ensemble, Wigner surmise
We investigate the eigenvalue spacing distributions for randomly
generated 4-regular Cayley graphs on SL2 (Fp)
by numerically calculating their spectra. We present strong evidence
that the distributions are Poisson and hence do not follow the Gaussian
orthogonal ensemble. Among the Cayley graphs of
SL2 (Fp) we consider are the new expander graphs
recently discovered by Y. Shalom.
In addition, we use a Markov chain method to generate random 4-regular
graphs, and observe that the average eigenvalue spacings are closely
approximated by the Wigner surmise.
*Departments of Mathematics and Computer Science, Dartmouth College,
Hanover, NH 03755