Quantum Chaos on Random Cayley Graphs of SL2[Z/pZ]

Igor Rivin; Naser T. Sardari

doi:10.1080/10586458.2017.1403982

Quantum Chaos on Random Cayley Graphs of SL₂[Z/_pZ]

Igor Rivin, Naser T. Sardari

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL₂[Z/_pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i.e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL₂[Z/_pZ] and the explicit LPS Ramanujan projective graphs of P¹(Z/_pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

Original language	English (US)
Pages (from-to)	328-341
Number of pages	14
Journal	Experimental Mathematics
Volume	28
Issue number	3
DOIs	https://doi.org/10.1080/10586458.2017.1403982
State	Published - Jul 3 2019

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1080/10586458.2017.1403982

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title = "Quantum Chaos on Random Cayley Graphs of SL2[Z/pZ]",

abstract = "We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i.e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.",

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AU - Sardari, Naser T.

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N2 - We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i.e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

AB - We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs, i.e., the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL2[Z/pZ] and the explicit LPS Ramanujan projective graphs of P1(Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

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Quantum Chaos on Random Cayley Graphs of SL₂[Z/_pZ]

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