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) [ ]

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

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

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

? Is there any physics Applied Mathematics 38 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres 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) Applied Mathematics 37 Statistics of Real Eigenvalues in GinOE Spectra [ ] » Ginibres random matrices: also physics

? 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

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

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

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

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

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

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

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

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

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

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

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

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…

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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