Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 6 / 20
Extremal eigenvalues & Tracy–Widom Law Statistics of extremal cases - the largest eigenvalue xmax The normalized largest eigenvalue s := (xmax − 2 √ N)N −1/6 of a GUE random matrix is (asymptotically) distributed according to the Tracy-Widom law (1994) F2(s) = det(1 − K) , where K is the integral operator with the Airy kernel K(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) . x − y The scaling behaviour of the finite size effect (as 1/N 1/6 ) is due to Bowick & Brezin (1991) and Forrester (1991). K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 7 / 20
Page 1 and 2: Level Spacing Distribution Revisite
Page 3 and 4: Random Matrices & Universality Univ
Page 5: Wigner Semicircle Law Spectral dens
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:

Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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