Level Spacing Distribution Revisited - Theoretical Group Atomic ...
Level Spacing Distribution Revisited - Theoretical Group Atomic ... Level Spacing Distribution Revisited - Theoretical Group Atomic ...
Gaussian Ensembles of Hermitian matrices H Hermitian random matrix H = H † consists of independent Gaussian entries a) orthogonal ensemble, β = 1 - real random numbers, b) unitary ensemble, β = 2 - complex numbers (real at the diagonal!) c) symplectic ensemble, β = 4 - quaternions (2 × 2 matrices) leading to 2N × 2N matrix with each eigenvalue occurring twice. Different normalization conditions, we use the one implied by the normal distribution 〈Hij〉 = 0 and σ2 = 〈H2 ij 〉 = 1 GUE ⇒ Unitary invariance, P(H) = P(UHU † ) leads to joint probability distribution (jpd) of eigenvalues xi β P − Pβ(x1, . . .,xN) = CNe 2 j x2 j |xj − xk| β j
Wigner Semicircle Law Spectral density P(x) can be obtained by integrating out all eigenvalues but one from jpd. For all three Gaussian ensembles of Hermitian random matrices one obtains (asymptotically, for N → ∞) the Wigner Semicircle Law (1955) P(x) = 1 2 − x2 2π where x denotes a normalized eigenvalue, xi = λi/ √ N K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 5 / 20
- Page 1 and 2: Level Spacing Distribution Revisite
- Page 3: Random Matrices & Universality Univ
- 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:
Wigner Semicircle Law<br />
Spectral density P(x)<br />
can be obtained by integrating out all eigenvalues but one from jpd.<br />
For all three Gaussian ensembles of Hermitian random matrices one<br />
obtains (asymptotically, for N → ∞) the Wigner Semicircle Law (1955)<br />
P(x) = 1 <br />
2 − x2 2π<br />
where x denotes a normalized eigenvalue, xi = λi/ √ N<br />
K ˙Z (IF UJ/CFT PAN ) <strong>Level</strong> <strong>Spacing</strong> <strong>Distribution</strong> March 14, 2011 5 / 20
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