Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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Extremal spacings distributions for random unitary matrices consider a) Minimal spacing smin = minj{sj} N j=1 and b) Maximal spacing smax = maxj{sj} N j=1 Minimal spacing distribution for N = 4 random unitary matrices Two qubits & random local gates Analytical results P2⊗2(t) for CUE(2) ⊗ CUE(2) case, where t = smin P2⊗2(t) = 1 2π(1−t) 4π 4−cos( πt 2 )−3 sin( πt 2 )+8 sin(πt)−3 sin(3πt 2 ) CUE, β = 2, CUE(4), P (2) 4 (t) = ... explicit result to long to reproduce it here... Poisson ensemble, β = 0, CPE(4), P (0) 4 (t) = 3(1 − t)2 . K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 16 / 20
Minimal spacing P(smin) for N = 4 unitary matrices Comparison of spacing distribution P(smin) for a) Poisson CPE(4), b) CUE(2) ⊗ CUE(2), c) CUE(4). mean values: 〈smin〉CPE4 = 1/4, 〈smin〉CUE2⊗CUE2 ≈ 0.4, 〈smin〉CUE4 ≈ 0.54 K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 17 / 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 and 12: Random Domino & Arctic Circle Theor
Page 13 and 14: Level spacing distribution P(s) Nea
Page 15: Level spacing for unitary matrices
Page 19 and 20: Average maximal spacing 〈smax〉
Page 21: Concluding Remarks Random Matrices:

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