A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

Journal of Statistical Mechanics: Theory and Experiment

Volumn 2011, Issue 4, 2011, Pages

  Nadal, Celine ^a   Majumdar, Satya N ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Painlev equations; random matrix theory and extensions

Indexed keywords

EID: 79955848621     PISSN: 17425468     EISSN: 17425468     Source Type: Journal    
DOI: 10.1088/1742-5468/2011/04/P04001     Document Type: Article

References (50)

1. Wigner E P 1951 On the statistical distribution of the widths and spacings of nuclear resonance levels Math. Proc. Camb. Phil. Soc. 47 790
2. Mehta M L 1991 Random Matrices 2nd edn (New York: Academic)
3. Dyson F J 1962 The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics J. Math. Phys. 3 1199
4. Rosenzweig N and Porter C E 1960 Repulsion of energy levels in complex atomic spectra Phys. Rev. 120 1698
5. Gumbel E J 1958 Statistics of Extremes (New York: Columbia University Press)
6. Tracy C A and Widom H 1994 Level spacing distributions and the Airy kernel Commun. Math. Phys. 159 151 [arXiv:hep-th/9211141] (Pubitemid 24759200)
7. Tracy C A and Widom H 1996 On orthogonal and symplectic matrix ensembles Commun. Math. Phys. 177 727 [arXiv:solv-int/9509007]
8. Forrester P J 1993 The spectrum edge of random matrix ensembles Nucl. Phys. B 402 709
9. Hastings S P and McLeod J B 1980 A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation Arch. Ration. Mech. Anal. 73 31
10. Aldous D and Diaconis P 1999 Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem Bull. Am. Math. Soc. 36 413
11. Majumdar S N 2007 Random matrices, the Ulam problem, directed polymers and growth models, and sequence matching Complex Systems (Les Houches Lecture Notes) ed J-P Bouchaud, M Mézard and J Dalibard (Amsterdam: Elsevier) pp 179-216
12. Forrester P J 2010 Log-Gases and Random Matrices (Princeton, NJ: Princeton University Press)
13. Kriecherbauer T and Krug J 2010 A pedestrian's view on interacting particle systems, KPZ universality and random matrices J. Phys. A: Math. Theor. 43 403001
14. Johansson K 1998 The longest increasing subsequence in a random permutation and a unitary random matrix model Math. Res. Lett. 5 63
15. Baik J, Deift P and Johansson K 1999 On the distribution of the length of the longest increasing subsequence of random permutations J. Am. Math. Soc. 12 1119
16. Prähofer M and Spohn H 2000 Universal distributions for growth processes in 1 + 1 dimensions and random matrices Phys. Rev. Lett. 84 4882
17. Gravner J, Tracy C A and Widom H 2001 Limit theorems for height fluctuations in a class of discrete space and time growth models J. Stat. Phys. 102 1085
18. Johansson K 2003 Discrete polynuclear growth and determinantal processes Commun. Math. Phys. 242 277
19. Majumdar S N and Nechaev S 2004 Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy-Widom distribution Phys. Rev. E 69 011103
20. Rambeau J and Schehr G 2010 Extremal statistics of curved growing interfaces in 1 + 1 dimensions Europhys. Lett. 91 60006
21. Johansson K 2000 Shape fluctuations and random matrices Commun. Math. Phys. 209 437
22. Sasamoto T and Spohn H 2010 One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality Phys. Rev. Lett. 104 230602
23. Calabrese P, Le Doussal P and Rosso A 2010 Free-energy distribution of the directed polymer at high temperature Europhys. Lett. 90 20002
24. Dotsenko V 2010 Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers Europhys. Lett. 90 20003
25. Amir G, Corwin I and Quastel J 2011 Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions Commun. Pure Appl. Math. 64 466
26. Majumdar S N and Nechaev S 2005 Exact asymptotic results for the Bernoulli matching model of sequence alignment Phys. Rev. E 72 020901
27. Forrester P J, Majumdar S N and Schehr G 2011 Non-intersecting Brownian walkers and Yang-Mills theory on the sphere Nucl. Phys. B 844 500
28. Rambeau J and Schehr G 2011 Distribution of the time at which N vicious walkers reach their maximal height arXiv:1102.1640
29. Miettinen L et al 2005 Experimental determination of KPZ height-fluctuation distributions Eur. Phys. J. B 46 55
30. Takeuchi K A and Sano M 2010 Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals Phys. Rev. Lett. 104 230601
31. Fridman M et al 2010 Measuring maximal eigenvalue distribution of Wishart random matrices with coupled lasers arXiv:1012:1282
32. Dean D S and Majumdar S N 2006 Large deviations of extreme eigenvalues of random matrices Phys. Rev. Lett. 97 160201 [arXiv:cond-mat/0609651]
33. Dean D S and Majumdar S N 2008 Extreme value statistics of eigenvalues of Gaussian random matrices Phys. Rev. E 77 041108 [arXiv:0801.1730]
34. Vivo P, Majumdar S N and Bohigas O 2007 Large deviations of the maximum eigenvalue in Wishart random matrices J. Phys. A: Math. Theor. 40 4317 [arXiv:cond-mat/0701371]
35. Majumdar S N and Vergassola M 2009 Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices Phys. Rev. Lett. 102 060601 [arXiv:0811.2290]
36. Ben Arous G, Dembo A and Guionnet A 2001 Aging of spherical spin glasses Probab. Theory Relat. Fields 120 1
37. Borot G, Eynard B, Majumdar S N and Nadal C 2010 Large deviations of the maximal eigenvalue of random matrices arXiv:1009.1945v3
38. Baik J, Buckingham R and DiFranco J 2007 Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function Commun. Math. Phys. 280 463
39. Gross D J and Matytsin A 1994 Instanton induced large N phase transitions in two and four dimensional QCD Nucl. Phys. B 429 50 [arXiv:hep-th/9404004]
40. Migdal A A 1975 Recursion equations in gauge field theories Sov. Phys. JETP 42 413
41. Rusakov B 1990 Loop averages and partition functions in U(N) gauge theory on two-dimensional manifolds Mod. Phys. Lett. A 5 693
42. 2 Phys. Lett. B 319 219
43. Crescimanno M, Naculich S G and Schnitzer H J 1996 Evaluation of the free energy of two-dimensional Yang-Mills theory Phys. Rev. D 54 1809 [arXiv:hep-th/9601104]
44. Gross D J and Witten E 1980 Possible third-order phase transition in the large-N lattice gauge theory Phys. Rev. D 21 446
45. Wadia S R 1980 phase transition in a class of exactly soluble model lattice gauge theories Phys. Lett. B 93 403
46. Periwal V and Shevitz D 1990 Unitary-matrix models as exactly solvable string theories Phys. Rev. Lett. 64 1326
47. Mariño M 2008 Nonperturbative effects and nonperturbative definitions in matrix models and topological strings J. High Energy Phys. JHEP12(2008)114
48. Schehr G, Majumdar S N, Comtet A and Randon-Furling J 2008 Exact distribution of the maximal height of p vicious walkers Phys. Rev. Lett. 101 150601
49. Forrester P J and Ormerod C M 2010 Differential equations for deformed Laguerre polynomials J. Approx. Theory 162 653
50. Chen Y and Feigin M V 2006 Painlevé IV and degenerate Gaussian unitary ensembles J. Phys. A: Math. Gen. 39 12381