Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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Tracy–Widom distributions Tracy–Widom distributions Fβ(s) Distributions Fβ(s) and the largest eigenvalue of random GUE matrices (image by A. Edelman) K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 8 / 20
Some applications of Tracy–Widom Law Combinatorics: length of the longest increasing permutation Baik, Deift, Johansson (1999) Statistics: largest eigenvalues of correlation matrices C = XX T Johnstone (2001) Growth processes: fluctuations around the limiting shape Johansson (2000) Superconductors: fluctuations of excitation gap in a quantum dot Vavilov, Brouer, Ambegaokar, Benakker (2001) Random Tilings & Aztec Diamond: fluctuations around the Arctic Circle Johansson (2002) K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 9 / 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: Extremal eigenvalues & Tracy-Widom
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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