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Statistics of Real Eigenvalues in GinOE Spectra Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007 Eugene Kanzieper Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel Gernot Akemann (Brunel) Phys. Rev. Lett. 95, (2005) arXiv: math-ph/ (J. Stat. Phys.) Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra 42 in preparation Alexei Borodin (Caltech) [ ]

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What is the probability that an n × n random real matrix with Gaussian i.i.d. entries has exactly k real eigenvalues? A. Edelman (mid-nineties) Statistics of Complex Spectra Applied Mathematics 41 Statistics of Real Eigenvalues in GinOE Spectra [ ] » The Problem

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Ginibres random matrices Definitions & physics applications Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Applied Mathematics 40 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline Pfaffian integration theorem

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1965 complexity s u c c e s s Statistics of Complex Spectra Is there any physics Dropped Hermiticity… ? Applied Mathematics 39 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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? Is there any physics Applied Mathematics 38 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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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) Applied Mathematics 37 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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? Is there any physics << 1 GinOE model ~ 1 directed chaos Applied Mathematics 36 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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Applied Mathematics Universal noise dressing is still there ! Asymmetric L-R Cross-Correlation Matrices 35 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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Applied Mathematics 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 Ginibres maths curiosity… 34 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

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Applied Mathematics 33 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Pfaffian integration theorem Ginibres random matrices Definitions & physics applications

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1965 GinUE GinSE (almost) uniform distribution depletion from real axis accumulation along real axis Applied Mathematics 32 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices GinOE

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1965 GinUE GinSE GinOE (almost) uniform distribution depletion from real axis accumulation along real axis Applied Mathematics 31 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices

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1965 GinUE (almost) uniform distribution GinUE : jpdf + correlations Applied Mathematics 30 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices

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1965 GinUE : jpdf + correlations GinSE depletion from real axis GinSE : jpdf + correlations Mehta, Srivastava 1966 Applied Mathematics 29 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices

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1965 GinUE : jpdf + correlations GinSE : jpdf + correlations Mehta, Srivastava 1966 GinOE accumulation along real axis Applied Mathematics 28 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices

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1965 GinOE accumulation along real axis … Key Feature 0 number of real eigenvalues 0 ? Applied Mathematics 27 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Spectra of Ginibres random matrices

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Applied Mathematics 26 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Pfaffian integration theorem Ginibres random matrices Definitions & physics applications

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1965 Ginibre 1991 Lehmann & Sommers 1997 Edelman 1994 Edelman, Kostlan & Shub quarter of a century !! Correlation Functions ?! 0 Applied Mathematics 25 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Overview of major developments since 1965

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Applied Mathematics 24 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Overview of major developments since Ginibre 1991 Lehmann & Sommers 1997 Edelman 1994 Edelman, Kostlan & Shub quarter of a century !! Correlation Functions ?! 0 Borodin & Sinclair, arXiv: Forrester & Nagao, arXiv: Sommers, arXiv: detailed k -th partial correlation functions are not available…

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Applied Mathematics 23 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Pfaffian integration theorem Ginibres random matrices Definitions & physics applications

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1997 Edelman Probability to have all eigenvalues real (the smallest one) Theorem ( rational) Applied Mathematics 22 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Real vs complex eigenvalues

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1997 Edelman + Solved ?.. Applied Mathematics 21 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Real vs complex eigenvalues

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MATHEMATICA code up to No Closed Formula for Applied Mathematics 20 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Real vs complex eigenvalues

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Applied Mathematics 19 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Pfaffian integration theorem Ginibres random matrices Definitions & physics applications

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Even Better Starting point The Answer a probability to have all eigenvalues real universal multivariate polynomials integer partitions a nonuniversal ingredient zonal polynomials Jack polynomials at α=2 Applied Mathematics 18 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Probability to find exactly k real eigenvalues

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No visible discrepancies with numeric simulations over 10 orders of magnitude !! Applied Mathematics 17 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Probability to find exactly k real eigenvalues

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Starting point GOE characteristic polynomial Nagao-Nishigaki (2001), Borodin-Strahov (2005) cancellation Reduced integral representation Applied Mathematics 16 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: I. Integrating out j s

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Reduced integral representation –part of a GOE matrix kernel GOE skew-orthogonal polynomials How do we calculate the integral ?.. not a projection operator ! Dyson Integration Theorem Inapplicable !! Applied Mathematics 15 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: I. Integrating out j s

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Applied Mathematics 14 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres real random matrices (GinOE) Overview of major developments since 1965 Real vs complex eigenvalues: What is (un)known ? Conclusions & What is next ? Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem Pfaffian integration theorem Ginibres random matrices Definitions & physics applications

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Applied Mathematics 13 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem Two fairly compact proofs

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Applied Mathematics 12 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem Apply !!

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a nonuniversal ingredient a probability to have all eigenvalues real Zonal polynomials Solved !! Applied Mathematics 11 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem

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Applied Mathematics 10 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem

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Applied Mathematics 09 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem Fredholm Pfaffian (Rains 2000)

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Applied Mathematics 08 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem

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Applied Mathematics 07 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem

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Applied Mathematics 06 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem

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Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra [ ] » Sketch of derivation: II. Pfaffian integration theorem 05

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Conclusions & What is next ? Applied Mathematics 04 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Outline reminder Ginibres 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 Ginibres random matrices Definitions & physics applications

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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 Pfaffian Integration Theorem as an extension of the Dyson Theorem (far beyond the present context) Applied Mathematics 03 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Conclusions

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Asymptotic analysis of the distribution of k (matrix size n taken to infinity) Further extension of the Pfaffian integration theorem to determine all partial correlation functions Looking for specific physical applications (weak non-Hermiticity) ! << 1 GinOE model ~ 1 directed chaos ? Applied Mathematics work in progress 02 Statistics of Real Eigenvalues in GinOE Spectra [ ] » What is next ? Asymptotic analysis of the distribution of k (when k scales with E[ k ] and the matrix size n that is taken to infinity)

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Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra 01 Statistics of Real Eigenvalues in GinOE Spectra Eugene Kanzieper Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel Gernot Akemann (Brunel) Phys. Rev. Lett. 95, (2005) arXiv: math-ph/ (J. Stat. Phys.) in preparation Alexei Borodin (Caltech) [ ] Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007

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