Level Spacing Distribution Revisited - Theoretical Group Atomic ...

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Aztec Diamond & domino lattice Aztec diamond of order n – a region of 2n(n + 1) unit squares tiled with n(n + 1) dominos (i.e. union of two squares) The Aztec diamond of order n = 3 K(3) = 2 6 = 64 There exists exactly K = 2 n(n+1)/2 different domino tilings Elkies, Kuperberg, Larsen, Propp (1992) How a random Aztec Diamond looks like?? K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 10 / 20
Random Domino & Arctic Circle Theorem Arctic Circle theorem - randomness inside the circle & ’Frozen Phase’ in four corners Jockush, Propp, Shor (1995) n = 50, K = 2 1275 ≈ 10 383 Fluctuations around the Arctic Circle are described by the Tracy–Widom distribution F2 Johansson (2002) K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 11 / 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: Some applications of Tracy-Widom La
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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