Computer Science Department
School of Computer Science, Carnegie Mellon University


Level Spacings for SL(2,p)

John D. Lafferty, Daniel N. Rockmore*

January 1997*

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.

17 pages

*Departments of Mathematics and Computer Science, Dartmouth College, Hanover, NH 03755

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