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[ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra ] Statistics of Real Eigenvalues in GinOE Spectra Eugene Kanzieper Gernot Akemann Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel (Brunel) Phys. Rev. Lett. 95, 230501 (2005) arXiv: math-ph/0703019 (J. Stat. Phys.) Alexei Borodin (Caltech) in preparation Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007 42 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » The Problem 2005 2007 What is the probabilityStatistics that an n × of n random real matrix with Gaussian i.i.d. entries has exactly k real eigenvalues? Spectra A.Complex Edelman (mid-nineties) 41 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 40 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] 39 » Ginibre’s random matrices: also physics P (H)DH N / 2 exp tr HH dHij( q ) GinOE : H R N N ( 1) GOE GinSE : H QN N ( 4) GSE GinUE : H CN N ( 2) GUE N q 1 i , j 1 1965 success complexity ( ) N 2 Dropped Hermiticity… Statistics of Is there any Complex Spectra physics ? Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Ginibre’s random matrices: also physics Is there any physics ? 38 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra » Ginibre’s random matrices: also physics • Dissipative quantum chaos (Grobe and Haake 1989) • Dynamics of neural networks (Sompolinsky et al 1988, Timme et al 2002, 2004) • Disordered systems with a direction (Efetov 1997) • QCD at a nonzero chemical potential (Stephanov 1996) • Integrable structure of conformal maps (Mineev-Weinstein et al 2000) • Interface dynamics at classical and quantum scales (Agam et al 2002) • Time series analysis of the brain auditory response (Kwapien et al 2000) • More to come: Financial correlations in stock markets (Kwapien et al 2006) ] 37 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Ginibre’s random matrices: also physics directed chaos << 1 ? Is there any HL HS h g H A physics GinOE model ~1 36 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] 35 » Ginibre’s random matrices: also physics Asymmetric L-R Cross-Correlation Matrices Universal noise dressing is still there ! Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Ginibre’s random matrices: also physics • Dissipative quantum chaos (Grobe and Haake 1989) • Dynamics of neural networks (Sompolinsky et al 1988, Timme et al 2002, 2004) • Disordered systems with a direction (Efetov 1997) • QCD at a nonzero chemical potential (Stephanov 1996) • Integrable structure of conformal maps (Mineev-Weinstein et al 2000) • Interface dynamics at classical and quantum scales (Agam et al 2002) • Time series analysis of the brain auditory response (Kwapien et al 2000) • More to come: Financial correlations in stock markets (Kwapien et al 2006) Back to 1965 and Ginibre’s maths curiosity… 34 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 33 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices ( ) N P (H)DH (almost) uniform distribution N / 2 2 exp tr HH dHij( q ) N q 1 i , j 1 depletion from real axis 1965 accumulation along real axis 32 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices 1965 (almost) uniform distribution depletion from real axis accumulation along real axis 31 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices N PN z1 ,..., z N C2 ( N ) zk1 zk2 k1 k2 2 N zk zk e 1965 k 1 GinUE : jpdf + correlations (almost) uniform distribution 30 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices N PN z1 ,..., zN C4 ( N ) zk1 zk2 zk1 zk2 k1 k2 2 2 N zk zk z z e k k k 1 2 1965 Mehta, Srivastava 1966 GinUE : jpdf + correlations GinSE : jpdf + correlations depletion from real axis 29 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices N PH T ( N / N ) ( w 1 ,, w N ) j1 j2 j1 j2 N e j 1 j2 / 2 1965 Mehta, Srivastava 1966 GinUE : jpdf + correlations GinSE : jpdf + correlations accumulation along real axis 28 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Spectra of Ginibre’s random matrices NN N k j10 1 j2 j 1 PHHTT((NN/)N()w( w , , w ) PH Tj1( N/ k) (j2w 1 ,e, w N ) 1 ,1 , w N N) j2 / 2 ? 1965 Key Feature H T (N ) T (N ) … N accumulation along T ( Nreal ) axis T ( N k 10 / k) T ( N / 0) ... T ( N / k ) ... T ( N / N ) number of real eigenvalues 27 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 26 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra 25 ] » Overview of major developments since 1965 Ginibre 1965 Lehmann & Sommers Edelman, Kostlan & Shub Edelman 1994 1997 1991 quarter of a century !! NN 2n 1 1 E [ k ] k p 1 O ) , wPNH,kT{( N /, PHT ( N / N ) ({wP}) H T ( N ) ( w 1 , , w N k ) ( w 1 , n, w N )} w w 1k, 0 N 1 , k , z 1 , z 1 ,, z l , z l2 k 0 1 k real eigenvalues l pairs of c.c. eigenvalues Correlation Functions ?! N PH T ( N / k ) ({w}) C N , k wi w j i j N w 2 e j erfc w j w j 2 j 1 1/ 2 [ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra ] » Overview of major developments since 1965 Ginibre Lehmann & Sommers 1965 Edelman, Kostlan & Shub Edelman 1994 1997 1991 quarter of a century !! N PH T ( N ) ( w 1 ,, w N ) PH T ( N / k ) ( w 1 ,, w N ) k 0 1 Correlation Functions ?! Borodin & Sinclair, arXiv: 0706.2670 Forrester & Nagao, arXiv: 0706.2020 Sommers, arXiv: 0706.1671 detailed k-th partial correlation functions are not available… 24 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 23 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Real vs complex eigenvalues Edelman N p N ,k dw j PH T ( N / k ) ({w}) 1997 j 1 Probability to have all eigenvalues real pN ,N 2 N ( N 1) / 4 (the smallest one) Theorem pN ,k rN ,k sN ,k 2 ( rN ,k & s N ,k rational) 22 Applied Mathematics [ ] Statistics of Real Eigenvalues in GinOE Spectra » Real vs complex eigenvalues p N ,k l 2 2 cN ,k k zp zp z p z p ( zEdelman N j2 / 2 2 p zp ) / 2 d Z p erfc e d j e k!l ! pj 1 2 i i 2 1 ({w}) dw Im{ Z }0 pP N ,k k j 1 i j i j N k 2l j l z pq p H T ( N / k ) zq 2 z p zq 2 k 1997 l j 1 p 1 j z p j z p wp w 1, ,2 wNN(N{1) 1/ ,4, k , z 1, z 1,, z l , z l } pN,k rN ,k N ,N k real eigenvalues l pairs of c.c. eigenvalues Solved ?.. s N ,k 2 + 21 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Real vs complex eigenvalues MATHEMATICA code up to N 9 No Closed Formula for p N ,k pN , N 2 N ( N 1) / 4 20 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 19 [ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra ] 18 » Probability to find exactly k real eigenvalues The Answer a probability to have all eigenvalues real pN ,k pN , N Fl ( x1 ,, xl ) N k 2l universal multivariate polynomials j 1 l 2xl j z p z p x j z ptr( 0z,p[ N / 2](z1) zˆ) / 2j c N ,k l / 2 Fl p(Nx,k1 ,, xl ) ( d1)j e d Z p erfc e ! 1 l j 2i k!l ! j 1 {l } jIm{ 1 Z i 2 }0j p k p (l ) g 2 j 2 p 2 p a nonuniversal ingredient k integer partitions i j i j l z pq p zq Even Better l l 1 1 , l 2 2 ,, l gGg N ( z ) Starting point 2 z p zq [ N / 2] 2 1 Fl l ( x1 ,..., x ) l zZ(1l ) ( x1 ,..., xl ) z j p j l !p j 1 p 1 k l z p N , N det 1̂ z ˆ polynomials pN , N 2l zonal l 0 Jack polynomials at α=2 Applied Mathematics [ 17 ] Statistics of Real Eigenvalues in GinOE Spectra » Probability to find exactly k real eigenvalues 0.01 0.0001 1. 10 6 1. 10 8 1. 10 10 0 2 4 6 8 10 12 8 10 12 0.4 0.3 0.2 0.1 0 0 2 4 6 No visible discrepancies with numeric simulations over 10 orders of magnitude !! Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] 16 » Sketch of derivation: I. Integrating out j ‘ s cancellation Reduced integral representation ({z, z}) 2l 1 l pf K N ({z, z}) z p z p ( z p2 z p2 ) / 2 2l 22l ( k kK ) characteristic GOE pf polynomial 2l ({pzN,,kz}) e A N ,l d Z p erfc N ({ z , z }) 2l 2l i 2 Im{ Z } 0 p 1 l det z Nagao-Nishigaki (2001), Borodin-Strahov (2005) j 1 Starting point p N ,k j Oˆ det z j Oˆ l cN ,k k zp zp z p z p ( z p2 z p2 ) / 2 j2 / 2 2 d e d Z erfc e j p k!l ! j 1 2i i 2 Im{ Z }0 p 1 k i j i j l z pq p zq 2 z p zq 2 k l j 1 p 1 j z p j z p Oˆ GOE( k k ) Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] 15 » Sketch of derivation: I. Integrating out j ‘ s Reduced integral representation pN ,k A N ,l S I K T D S z p z p ( z p2 z p2 ) / 2 e d Z p erfc pf K N ({z, z}) 2l 2l i 2 Im{ Z } 0 p 1 l 2 D –part of a GOE 2 2 matrix kernel GOE skew-orthogonal polynomials not a projection operator ! Dyson Integration Theorem Inapplicable !! How do we calculate the integral ?.. Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 14 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem Two fairly compact proofs 13 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem p N ,k A N ,l z p z p ( z p2 z p2 ) / 2 e d Z p erfc pf K N ({z, z}) 2l 2l i 2 Im{ Z } 0 p 1 l 2 12 [ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra 11 ] » Sketch of derivation: II. Pfaffian integration theorem pN ,k pN , N l! a probability to have all eigenvalues real Z (1l ) ( x1 ,, xl ) N k 2l pN , N 2 N ( N 1) / 4 Zonal polynomials GN ( z ) [ N / 2] z l 0 l pN , N 2l pN , N det 1̂ z ˆ x j tr (0, [ N / 2]1) ˆ j a nonuniversal ingredient Solved !! Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem 10 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] 09 » Sketch of derivation: II. Pfaffian integration theorem Fredholm Pfaffian (Rains 2000) Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem 08 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem 07 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem 06 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Sketch of derivation: II. Pfaffian integration theorem 05 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » Outline reminder Ginibre’s random matrices • Definitions & physics applications Ginibre’s real random matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ? 04 [ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra ] » Conclusions Statistics of real eigenvalues in GinOE Exact formula for the distribution of the number k of real eigenvalues in the spectrum of n × n random Gaussian real (asymmetric) matrix Solution highlights a link between integrable structure of GinOE and the theory of symmetric functions Even simpler solution is found for the entire generating function of the distribution of k 0.01 0.0001 Pfaffian Integration Theorem as an extension of the 1. 10 6 1. 10 8 Dyson Theorem (far beyond the present context) 1. 10 10 0 2 4 6 8 10 12 03 Applied Mathematics [ Statistics of Real Eigenvalues in GinOE Spectra ] » What is next ? Looking for specific physical applications (weak non-Hermiticity) ! directed chaos << 1 HL HS h g H A GinOE model ~1 ? Asymptotic analysis of the distribution of k (matrix size n taken to infinity) work in progress Asymptotic analysis of the distribution of k (when k scales with E[k] and the matrix size n that is taken to infinity) Further extension of the Pfaffian integration theorem to determine all partial correlation functions 02 [ Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra ] Statistics of Real Eigenvalues in GinOE Spectra Eugene Kanzieper Gernot Akemann Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel (Brunel) Phys. Rev. Lett. 95, 230501 (2005) arXiv: math-ph/0703019 (J. Stat. Phys.) Alexei Borodin (Caltech) in preparation Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007 01