Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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Random matrices & Finite N Effects Analogy Infinite matrices ↔ semiclassical → 0 limit ↔ geometric optics diffraction and Airy discs Finite N Random Matrices ↔ Quantum Theory > 0 ↔ Wave Optics K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 12 / 20
Level spacing distribution P(s) Nearest neighbour spacing s si = xi+1−xi ∆ where ∆ is the mean spacing a) Gaussian ensembles for N = 2 ⇒ Wigner surmise β = 1 GOE (orthogonal) P1(s) = π 2s exp(−π 4s2 ) β = 2 GUE (unitary) P2(s) = 32 π 2s 2 exp(− 4 π s2 ) β = 4 GSE (symplectic) P4(s) = 218 3 6 π 3s 4 exp(− 64 9π s2 ) These distributions derived for N = 2 work well also for Gaussian ensembles in the asymptotic case, N → ∞. Random unitary matrices & Circular ensembles of Dyson Uniform density of phases along the unit circle, P(φ) = 1/2π. Phase spacing, si = N 2π [φi+1 − φi] since ∆ = 2π/N. For large matrices the level spacing distributions for Gaussian ensembles (Hermitian matrices) and circular ensembles (unitary matrices) coincide. K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 13 / 20
Page 1 and 2: Level Spacing Distribution Revisite
Page 3 and 4: Random Matrices & Universality Univ
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: Random Domino & Arctic Circle Theor
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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