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Periodic walks on random graphs and random matrix theory

Tuesday 27th July 2010 - 09:00 to 10:00
INI Seminar Room 1
The spectral statistics of the discrete Laplacian of d-regular graphs on V vertices are intimately connected with the distribution of the number of cycles of period t (t-cycles) on the graph. I shall discuss this connection by using a trace formula which expresses the spectral density in terms of the t-cycle counts. The trace formula will be used to write the spectral pair correlations in terms of the properly normalized variance of the t-cycle counts. Based on these results, I would like to propose a conjecture which uses Random Matrix Theory to compute the variance of the t-periodic cycle counts in the limit V,t -> infinity fixed value of with t/V. Numerical computations support this conjecture.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons