Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs II - Mathematical PhysicsReport as inadecuate




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Abstract: Following the derivation of the trace formulae in the first paper in thisseries, we establish here a connection between the spectral statistics ofrandom regular graphs and the predictions of Random Matrix Theory RMT. Thisfollows from the known Poisson distribution of cycle counts in regular graphs,in the limit that the cycle periods are kept constant and the number ofvertices increases indefinitely. The result is analogous to the so called-diagonal approximation- in Quantum Chaos. We also show that by assuming thatthe spectral correlations are given by RMT to all orders, we can compute theleading deviations from the Poisson distribution for cycle counts. We providenumerical evidence which supports this conjecture.



Author: Idan Oren, Uzy Smilansky

Source: https://arxiv.org/



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