Level statistics of the correlated and uncorrelated Wishart ensemble. Mario Kieburg

Transcription

1 Level statistics of the correlated and uncorrelated Wishart ensemble Mario Kieburg Fakultät für Physik Universität Bielefeld (Germany) X Brunel-Bielefeld Workshop on Random Matrix Theory December 13th, 2014

2 My recent Collaborators Gernot Akemann Rene Wegner Thomas Guhr Tim Wirtz

3 Wishart matrices in real life Wishart matrix: WW, W R N 1 N 2, C N 1 N 2, H N 1 N 2 Lattice QCD (image by Göckeler, Hehl, Rakow, Schäfer, Wettig (1998)) Financial Market (image by Bouchaud, Potters, Laloux (2005)) Climate Research Telecommunications

4 What do we know about uncorrelated Wishart matrices? P(W ) = exp[ tr WW ] Determinantal (β = 2) or Pfaffian (β = 1, 4) point processes? det(ww λ j 1) Yes: for averages of det(ww κ j 1) and k-point correlation functions Level density? Yes: Marcenko-Pastur distribution Local Statistics? Yes: Sine-kernel (bulk), Airy-kernel (soft edge) Tracy-Widom distribution, Bessel-kernel (hard edge)

5 What do we know about correlated Wishart matrices? P(W ) = exp[ tr C 1 2 WC 1 1 W ] Determinantal (β = 2) or Pfaffian (β = 1, 4) point processes? Yes: for the case β = 2, C 2 = 1 otherwise we don t know Level density? Yes: for the case β = 1 and C 2 = 1 and for β = 2 and C 1, C 2 1 otherwise we didn t know Local Statistics? Yes: for the case β = 2 and C 2 = 1 otherwise we don t know (problems partially circumvented)

6 Outline of this talk Distribution of the smallest eigenvalue (uncorrelated) Correlated Wishart matrices Outlook: What is with non-gaussian ensembles?

7 Distribution of the smallest eigenvalue (uncorrelated) Smallest painting of the world, Mona Lisa" on a hair by Georgia Institute of Technology (image from huffingtonpost.co.uk)

8 Distribution of the Smallest Eigenvalue E min (s) = s P(0, s) P(0, s) = d[w ]P(W )Θ[WW s1] Laguerre ensemble P(W ) det γ WW e tr WW /(2σ 2) : β = 1: E min (s) s (ν 1)/2 e Ns det (ν 1)/2 (WW + s1) W : (N 1) (N+2) β = 2: E min (s) s ν e Ns det ν (WW + s1) W : (N 1) (N+1) β = 4: E min (s) s 2ν+1 e Ns det ν+1/2 (WW + s1)det 3/2 WW 0 s W : 2(N 1) 2(N 1) Well known results for: β = 2 and ν arbitrary; β = 1 and ν odd

9 E min for β = 2 [ ] E min (s) s ν e Ns det L (b a 2) N+ν+a b ( s) E min for β = 1, odd ν (integer power of det) [ E min (s) s (9ν+1)/2 e Ns Pf (a b)l (2ν a b 4) N+a+b+1 ] ( 2s) L (α) n is the modified Laguerre polynomial in monic normalization Nagao, Forrester 98, for N with ν fixed by Damgaard, Nishigaki, Wettig 98, Damgaard, Nishigaki 00

10 E min for β = 1, even ν E min (s) s (ν 1)/2 e Ns The square root is a problem! (?) det ν/2 (WW + s1) det(ww + s1) W : (N 1) (N+2) Edelman s Approach 91 E min (s) = s (ν 1)/2 e Ns [ U(a ν, b ν, s)q ν (s) + U(a ν, b ν, s)q ν(s) ] Q ν (s) and Q ν(s) polynomials given via a recursion U(a, b, x) is Tricomi s confluent hypergeometric function simple expressions for ν = 0, 2 Main idea: Recursion in ν!

11 E min for β = 1, even ν The square root is a problem! (?) E min (s) s (ν 1)/2 e Ns det ν/2 (WW + s1) det(ww + s1) s (ν+2)/2 e Ns det ν/2 (WW + s1) det(w W + s1) s (ν+2)/2 e Ns 1 det(w W + s1) W : (N 1) (N+2) W W det ν/2 (WW + s1) µ Combination of orthogonal polynomial theory and SUSY is the solution! Redefinition of the two-point weight: x 1 x 2 e x 1 x 2 d µ(x 1, x 2 ) = sign(x 1 x 2 ) (x1 + s)(x 2 + s) dx 1dx 2

12 De Bruijn Integral [ Z (2N) 2k (κ) = Pf ] 2k dµ(x a, x b ) 2N (x) det(x κ l 1 2N ) l=1 1 [ ] 2k (κ) Pf (κ a κ b )Z (2(N+k 1)) 2 (κ a, κ b ) 1 a,b 2k with antisymmetric two point measure: dµ(x 1, x 2 ) = dµ(x 2, x 1 )

13 Examples real Wishart (β = 1): dµ(x 1, x 2 ) = sign(x 1 x 2 )(x 1 x 2 ) (ν 1)/2 e x 1 x 2 dx 1 dx 2 quaternion Wishart (β = 4): dµ(x 1, x 2 ) = (x 1 x 2 ) 2ν+1 e x 1 x 2 ( x1 x2 )δ(x 1 x 2 )dx 1 dx 2 complex Wishart (β = 2): dµ(x 1, x 2 ) = (x N 1 x N 2 )2 x 1 x 2 (x 1 x 2 ) ν e x 1 x 2 dx 1 dx 2 Also the complex case fits into this framework! [ ] [ ] (x 2N (x) = ± det xa b 1 N = ±Pf a xb N)2 x a x b And our case: x 1 x 2 e x 1 x 2 d µ(x 1, x 2 ) = sign(x 1 x 2 ) (x1 + s)(x 2 + s) dx 1dx 2

14 E min for β = 1, even ν normalization constant: det 1/2 (W W + s1) W dxe sx x N/2 0 (1 + x) ( (N 1)/2 N + 2 U, 5 ) 2 2, s skew-orthogonal polynomials of even order 2j: det(y 1 WW ) µ = L (2) 2j (2y) 2j U ( j + 3 2, 3 2, s) U ( j + 3 2, 5 2, s)l(2) 2j 1 (2y) U(a, b, x) is Tricomi s confluent hypergeometric function

15 E min for β = 1, ν 4N Distribution of the smallest eigenvalue: ( N + 2 E min (s) s (ν+2)/2 e Ns U, 5 ) 2 2, s [ Pf s a 1 1 s b 1 2 (s 1 s 2 ) det(ww + s 1 1)det(WW + s 2 1) Two point partition function: either as a sum of skew-orthogonal polynomials or as W :j (j+3) det(ww + s 1 1)det(WW + s 2 1) µ 1 ( ) dx dµ Haar (U)e sx+tr diag(s 1,s 1,s 2,s 2 )U U j+3 2, 5 2, s 0 CSE(4) x (j+1)/2 det j/2 U det j/2+1 (1 + U)det 1/2 ((1 + x)1 + U) (1 + x) j/2+2 (Akemann, Guhr, Kieburg, Wegner, Wirtz 14 ) New result for finite ν and N! µ s1 =s 2 =s ]

16 Smallest P (λ min )= Eigenvalue 1 Distr. (β = 4) 2 λ min[2i 2 (2λ min ) I 0 (2λ min )+1]e 1 2 λ2 min. For the present re interesting. In forthechse[9] E min (s) s 2ν+1 e Ns det ν+1/2 (WW + s1)det 3/2 (6) WW N f +3 (λ 2 W n + m 2 f ) We shall also need P (λ min )forn f =2and4whichinthe f=1 absence of closed analytical expressions can perhaps most Microscopic easily be obtained universality from results with computed dynamical by Kaneko fermions [24]. (3) The essential ingredients are the zonal polynomials at the is the Vanderm-matrix 2 model symmetric M.E. Berbenni-Bitsch point. We obtain 1,S.Meyer after some 1,andT.Wettig algebra 2 1 Fachbereich Physik Theoretische Physik, Universität Kaiserslautern, D Kaiserslautern, Germany Institut für Theoretische Physik, Technische 2 Universität München, D Garching, Germany to account withit is intuitively P (λ min )= (α +1)!(α (September +3)! λ2α+3 24, min2013) e 1 2 λ2 min T (λ 2 min ), (7) eterminants It has recently will been demonstrated in quenched lattice simulations that the distribution of the lowlying eigenvalues where of the QCD T (x) Dirac =1+ operator d=1 s only if the m a isd x universal d with and described by random-matrix theory. f We present first evidence that this universality continues toholdinthepresenceofdynamical alue or, in other quarks. Data froma lattice simulation with gauge group SU(2) anddynamicalstaggeredfermions pacing are near compared zero d = α+2j i to the predictions of the chiral α+2j i+4 κ =d (i,j) κ symplectic ensemble of random-matrix theory with toobserveanefsses much implications largerof our results. 1 l(κ) α+1 massive dynamical quarks. Good agreement is found in this exploratory study. We also discuss eement with the [κ PACS numbers: Ha, b, j approximation. i+2(κ Rd, i j)+1][κ Gc j i+2(κ i j)+2]. (8) here the lattice Here, κ denotes a partition of the integer d, l(κ) its as conjectured a few years length, ago κ [1,2] itsthat weight, the miic spectral quanti- propertiesinof Eq. the(8), QCDa partition Dirac opera- κ is identified the effectwith of light itsdynamical diagram, quarks with masses of and κ will thebe conjugate addressed partition. in this work is the following. W spectral ical particular quarksthe have distribution κ = {s of the =(i, low-lying j); 1 eigenare universal ensem- and can series be for obtained T (x) israpidlyconvergent,andthecurvesfor in effective tor, and can this effect be described by results ob i l(κ), 1 1/(V j Σ) κonthelow-lyingspectrumofthedirac i }. The Taylor unitary sy that ensemble are much [18]. simpler N f =2and4caneasilybecomputedtoanydesired than QCD. This con- withon random- the observation accuracy. [3] that in the range with the analytical information available from in chiral RMT? The answer to these questions to tarests RMT

17 Correlated Wishart matrices The Three Wise Apes" (image from thomasbrasch-wordpress.com)

18 The RMT Model P(W ) exp[ tr C 1 WW ]/det γ(β)(n+ν) C Empirical correlation matrix: C = UΛU = Udiag(Λ 1,..., Λ N )U 0 (Bouchaud, Potters, Laloux (2005))

19 For β = 2 Harish-Chandra, Itzykson-Zuber integral: p(x) det ν x 2 N (x) dµ Haar (U) exp[ tr Λ 1 UxU ]/det N+ν Λ U (N) detν x N (x) det [ e x b/λ a ] det N+ν Λ N (Λ 1 ) The jpdf can be calculated! Everything is calculated!? level density (Alfano, [ Tulino, Lozano, Verdu 04 ): Λ b 2 ρ(y) det a Λ ν 1 a e y/λa y ν+b 1 /Γ[ν + b] 0 ] / N (Λ) gap probability for smallest [ eigenvalue ( (Forrester )] / 07 ): P(0, s) e tr s/λ det L (1 N ν b) s ν+b 1 N (s/λ) Λ a

20 No group integral for β = 1, 4 Harish-Chandra integral (known): dµ Haar (U) exp[ tr Λ 1 UxU ] G Λ 1 and x in the Lie algebra of the group G Itzykson-Zuber integral (unknown for β = 1, 4): dµ Haar (U) exp[ tr Λ 1 UxU ] G Λ 1 and x in a co-set corresponding to the group G β = 1: Λ 1 and x real symmetric β = 4: Λ 1 and x quaternion self-dual SUSY" circumvents group integrals!

21 For β = 1 level density (Recher, Kieburg, Guhr, Zirnbauer 10 ) Finite sum of products of two decoupling one-fold integrals gap probability for the smallest eigenvalue (Wirtz, Guhr 14 ) Yields always a (ν 1) (ν 1) or ν ν Pfaffian! But: The kernel depends on all Λ j level density: two-sided correlations (Waltner, Wirtz, Guhr 14 ) Finite sum of four-fold integrals Expressions drastically simplify when Λ is double degenerate!

22 For β = 1 Gap probability for the largest eigenvalue (Wirtz, Kieburg, Guhr 14 ): P(s, ) R N (N+ν) d[w ]e s tr Λ 1 WW Θ(1 W W ) det (N+ν)/2 (Λ/s) Ingham-Siegel integral = Heaviside Θ-function: P(s, ) 1 e ıtr H det (N+ν)/2 d[h] (Λ/s) Sym(N+ν) det (N+ν 1)/2 (ıh + 1) N 1 det(ıh (s/λj + 1)1) j=1 Yields always an (N + ν) (N + ν) Pfaffian! But: The kernel depends on all Λ j More details in the poster presentation by Tim Wirtz!

23 Outlook Outlook" by Peter Quidley (image from theoilpaintingsaless.com)

24 What is with non-gaussian ensembles? Laguerre: P(W ) = exp[ tr W C 1 W ] non-compact support all moments exist Jacobi: P(W ) = det γ (1 W C 1 W ) Θ(1 W C 1 W ) compact support all moments exist Lorentz (Cauchy): 1 P(W ) = det γ (1 + W C 1 W ) non-compact support not all moments exist Ρ Λ Gauss Jacobi Lorentz Λ (Bouchaud, Potters, Laloux (2005))

25 More Mathematical Challenges and Fun Meijer G-ensemble = Product of rectangular matrices Statistics of the Wishart matrix: WW = W 1 W 2 W M CW M W 2 W 1 What is the level density with a fixed correlation matrix C? What are the distributions of the smallest eigenvalues at the hard edge (also for C = 1)? New universality class! Meijer G-kernel What happens when the W j s are correlated with each other?

26 Thank you for your attention! C. Recher, M. Kieburg, T. Guhr: Phys. Rev. Lett. 105, (2010) [arxiv: ] C. Recher, M. Kieburg, T. Guhr, M. R. Zirnbauer: J. Stat. Phys. 148, 981 (2012) [arxiv: ] T. Wirtz, T. Guhr: Phys. Rev. Lett. 111, (2013) [arxiv: ] T. Wirtz, T. Guhr: J. Phys. A 47, (2014) [arxiv: ] G. Akemann, T. Guhr, M. Kieburg, R. Wegner, T. Wirtz: accepted for publication in Phys. Rev. Lett. [arxiv: ] T. Wirtz, M. Kieburg, T. Guhr: [arxiv: ] D. Waltner, T. Wirtz, T. Guhr: [arxiv: ]