Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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Quantum Chaos & Unitary Dynamics ’Quantum chaology’: analogues of classically chaotic systems Quantum analogues of classically chaotic dynamical systems can be described by random matrices a) autonomous systems – Hamiltonians: Gaussian ensembles of random Hermitian matrices, (GOE, GUE, GSE) b) periodic systems – evolution operators: Dyson circular ensembles of random unitary matrices, (COE, CUE, CSE) K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 2 / 20
Random Matrices & Universality Universality classes Depending on the symmetry properties of the system one uses ensembles form orthogonal (β = 1); unitary (β = 2) and symplectic (β = 4) ensembles. The exponent β determines the level repulsion, P(s) ∼ s β for s → 0 where s stands for the (normalised) level spacing, si = φi+1 − φi. see e.g. F. Haake, Quantum Signatures of Chaos K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 3 / 20
Page 1: Level Spacing Distribution Revisite
Page 5 and 6: Wigner Semicircle Law Spectral dens
Page 7 and 8: Extremal eigenvalues & Tracy-Widom
Page 9 and 10: Some applications of Tracy-Widom La
Page 11 and 12: Random Domino & Arctic Circle Theor
Page 13 and 14: Level spacing distribution P(s) Nea
Page 15 and 16: Level spacing for unitary matrices
Page 17 and 18: Minimal spacing P(smin) for N = 4 u
Page 19 and 20: Average maximal spacing 〈smax〉
Page 21: Concluding Remarks Random Matrices:

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