Near extreme eigenvalues and the first gap of Hermitian random matrices

Transcription

1 Near extreme eigenvalues and the first gap of Hermitian random matrices Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Sydney Random Matrix Theory Workshop January 2014 Anthony Perret, G. S., arxiv:

2 Extreme value statistics Statement of the problem X 1,X 2,,X N : N random variables, P joint (X 1,X 2,,X N ) X max = max 1appleiappleN X i,f N (M) =P(X max apple M) Q: N!1? Fully understood for i.i.d. random variables Three different universality classes depending on the pdf of X i : Gumbel, Fréchet, Weibull

3 Extreme value statistics Statement of the problem X 1,X 2,,X N : N random variables, P joint (X 1,X 2,,X N ) X max = max 1appleiappleN X i,f N (M) =P(X max apple M) Q: N!1? Very few exact results for strongly correlated variables Random walks Random matrices X max is interesting BUT concerns a single variable among N

4 Statistics of Near-Extremes Statistical Physics Energy levels see also: Branching Brownian motion T>0{ E 0 Ground state Natural sciences (e.g. seismology) T =0 1/f noise Brownian motion... Crowding near the extremes

5 Statistics of Near-Extremes How to quantify the crowding close to extreme values? th Look at (higher) order statistics: k maximum X max = M 1,N >M 2,N > >M N,N = X min and in particular the spacings (gaps) d k,n = M k,n M k+1,n Consider the density of near-extremes Sabhapandit, Majumdar 07 DOS (r, N)= 1 N 1 NX i=1,i6=imax h (X max X i r)i

6 Near-extreme eigenvalues of random matrices An interesting set of strongly correlated random variables P joint ( 1, 2,..., N )= 1 Y ( i j ) 2 P N e i=1 Z N i<j 2 i Wigner sea dr r Λ 2,N λ Λ 1,N = λ max st 1 gap Density of near extreme eigenvalues DOS (r, N)= 1 N 1 NX i=1 i6=imax Witte, Bornemann, Forrester 13 h ( max i r)i

7 Density of near extreme eigenvalues in GUE DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i Fluctuations of the largest eigenvalue in GUE max = p 2N + N 1/6 1 p 2 2 max = max 1appleiappleN i Tracy-Widom =2 Depending on (r, N) one expects two different regimes bulk regime edge regime

8 A detour by the density of eigenvalues of GUE random matrices Two regimes: bulk and edge regime

9 A detour by the density of eigenvalues of GUE random matrices Two regimes: bulk and edge regime Matching between the bulk and edge regimes matching with Wigner semi-circle coincides with the right tail of TW

10 Density of near extreme eigenvalues: results DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i In the bulk : A. Perret, G. S. 13 is insensitive to the fluctuations of shifted Wigner semi-circle

11 Density of near extreme eigenvalues: results DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i A. Perret, G. S. 13 At the edge : more complicated, and more interesting,

12 Density of near extreme eigenvalues: results is related to a solution of the Lax pair associated to Painlevé XXXIV

13 Density of near extreme eigenvalues: results Asymptotic behaviors 1 ρedge( r) 0 0 r 10

14 Near-extreme eigenvalues of random matrices Wigner sea dr r Density of near extreme eigenvalues DOS (r, N)= 1 N 1 NX Λ 2,N i=1 i6=imax λ Λ 1,N = λ max h ( max i r)i st 1 gap GAP (r, N)=P[( 1,N 2,N ) 2 [r, r + dr]] An exact identity A. Perret, G. S. 13 which relies on the invariance of the joint pdf of eigenvalues under the group of permutations

15 Typical fluctuations of the gap: results A. Perret, G. S. 13 Asymptotic behaviors

16 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

17 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

18 Orthogonal polynomials P joint ( 1, 2,..., N )= 1 Y ( i j ) 2 P N e i=1 Z N i<j 2 i Cumulative distribution function of Orthogonal polynomials on the semi-infinite real line C. Nadal, S. N. Majumdar 13

19 Recurrence relations for the norms Introduce a modified weight Recurrence relation for the polynomials Recurrence relation for the norms C. Nadal, S. N. Majumdar 11

20 Physical picture associated to the OP sytem Coulomb gas with a wall located in Moving the wall through the edge: the density y y y ρ y (λ) ρ y (λ) ρ y (λ) L(y) λ λ λ 2N 2N 2N 2N 2N y< 2N y = 2N y> 2N PUSHED CRITICAL PULLED described by a double scaling limit see also T. Claeys, A. Kuijlaars 08, «...when the soft edge meets the hard edge»

21 Recurrence relations for the norms and double scaling limit Double scaling limit keeping fixed Asymptotic behaviors of the norms in the double scaling limit C. Nadal, S. N. Majumdar 11 with yields an alternative derivation of TW distribution

22 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

23 An exact formula for DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i wall eigenvalues After some manipulations one obtains two-point correlations for conditioned eigenvalues wih the kernel

24 An exact formula for DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i wall eigenvalues Finally a useful formula is 2-point correlations

25 An exact formula for the PDF of the first gap wall eigenvalues Reminding that one gets

26 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

27 Large analysis of DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i Bulk regime : negligible to leading order Finally shifted Wigner semi-circle

28 Large analysis of DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i Edge regime : Analysis of the kernel in the double scaling limit where Find a solution of the recurrence in the double scaling limit

29 Large analysis of DOS (r, N)= 1 N 1 NX i=1 i6=imax h ( max i r)i Remember that, in the double scaling limit with In the double scaling limit the recurrence relation is solved by A. Perret, G. S. 13

30 Large analysis of After some more computations... where solve the Lax pair for Painlevé XXXIV see T. Claeys, A. Kuijlaars 07 for a (rigorous) derivation using RH

31 Density of near extreme eigenvalues: results is related to a solution of the Lax pair associated to Painlevé XXXIV

32 Typical fluctuations of the gap: results A. Perret, G. S. 13 using we obtain from which we obtain the asymptotic behaviors with and complicated integral involving

33 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

34 Relations with existing results Density of near extreme eigenvalues was not studied in RMT (to my knowledge) Previous studies of the PDF of the first gap an expression in terms of a Fredholm determinant Forrester 93 an expression in terms of Painlevé transcendents Witte, Bornemann, Forrester 13 and a numerical computation of the formula in terms of a Fredholm determinant

35 PDF of the first gap: the formula of Witte, Bornemann, Forrester with and with Show that this formula coincides with ours!

36 PDF of the first gap: the numerical evaluation of Witte, Bornemann, Forrester 0.5 ρtyp( r) 0 0 r 5

37 Outline Orthogonal polynomials (OPs) on the semi-infinite real line Exact formulas for for finite Asymptotic analysis for large Comparison with existing results Conclusion and related open problems

38 Conclusion and related open questions Exact results for the statistics of near extreme eigenvalues of GUE A new formula for the PDF of the first gap in terms of Painlevé transcendents Precise asymptotics th Similar formula for the k gap? What about GOE and GSE (skew orthogonal polynomials)? What about the hard edge?