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Statistics of Real Eigenvalues in GinOE Spectra

]. [. . 42. Applied Mathematics. Statistics of Real Eigenvalues in GinOE Spectra. Statistics of Real Eigenvalues in GinOE Spectra. Eugene Kanzieper. Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel. Alexei Borodin (Caltech).

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Statistics of Real Eigenvalues in GinOE Spectra

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  1. ] [  42 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra Statistics ofReal Eigenvalues in GinOE Spectra Eugene Kanzieper Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel Alexei Borodin (Caltech) Gernot Akemann (Brunel) Phys. Rev. Lett. 95, 230501 (2005) arXiv: math-ph/0703019 (J. Stat. Phys.) in preparation Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007

  2. ] [  41 Statistics of Complex Spectra Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » The Problem 2005 2007 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)

  3. ] [  40 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline  Ginibre’s random matrices •Definitions & physics applications Ginibre’srealrandom matrices (GinOE) • Overview ofmajordevelopments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and • inapplicabilityof the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ?

  4. ] [  39  complexity s u c c e s s Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Ginibre’s random matrices: also physics 1965 Dropped Hermiticity… Statistics of Complex Spectra ? Is there any physics

  5. ] [  38 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Ginibre’s random matrices: also physics ? Is there any physics

  6. ] [  37 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)

  7. ] [  36 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Ginibre’s random matrices: also physics << 1 directed chaos ? Is there any physics GinOE model ~ 1

  8. ] [  35 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Ginibre’s random matrices: also physics Asymmetric L-R Cross-Correlation Matrices Universal noise dressing is still there !

  9. ] [  34 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 to1965and Ginibre’s maths curiosity…

  10. ] [  33 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom matrices (GinOE) • Overview ofmajordevelopments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and • inapplicabilityof the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ?

  11. ] [  32  Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices 1965 GinSE GinOE GinUE (almost) uniform distribution depletion from real axis accumulation along real axis

  12. ] [  31 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices 1965 GinSE GinOE GinUE (almost) uniform distribution depletion from real axis accumulation along real axis

  13. ] [  30 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices 1965  : jpdf + correlations GinUE GinUE (almost) uniform distribution

  14. ] [  29 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices 1965  Mehta, Srivastava 1966 : jpdf + correlations GinUE : jpdf GinSE + correlations GinSE depletion from real axis

  15. ] [  28  Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices 1965 Mehta, Srivastava 1966 : jpdf + correlations GinUE : jpdf GinSE + correlations GinOE accumulation along real axis

  16. ] [  27 0  Key Feature 0  Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Spectra of Ginibre’s random matrices ? 1965 … GinOE accumulation along real axis number of real eigenvalues

  17. ] [  26 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom matrices (GinOE) • Overview ofmajordevelopments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and • inapplicabilityof the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ?

  18. ] [  25 0 Ginibre Edelman Edelman, Kostlan & Shub Lehmann & Sommers 1965 1994 1991 1997     Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Overview of major developments since 1965 quarter of a century !! Correlation Functions ?!

  19. ] [  24 0 Ginibre Edelman Edelman, Kostlan & Shub Lehmann & Sommers 1965 1994 1991 1997     quarter of a century !! Correlation Functions ?! Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Overview of major developments since 1965 Borodin & Sinclair, arXiv: 0706.2670 Forrester & Nagao, arXiv: 0706.2020 Sommers, arXiv: 0706.1671 detailed k-th partial correlation functions are not available…

  20. ] [  23 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and • inapplicabilityof the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ?

  21. ] [  22 Edelman 1997 (the smallest one)  ( rational) Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Real vs complex eigenvalues  Probability to have all eigenvalues real  Theorem

  22. ] [  21 Edelman 1997  + Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Real vs complex eigenvalues Solved ?..

  23. ] [  20 MATHEMATICA code up to NoClosed Formula for Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Real vs complex eigenvalues

  24. ] [  19 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom matrices (GinOE) • Overview of major developments since 1965 • Real vs complex eigenvalues: What is (un)known ? • Probability to find exactly k real eigenvalues and • inapplicabilityof the Dyson integration theorem • Pfaffian integration theorem Conclusions & What is next ?

  25. ] [  18 a probability to have all eigenvalues real universal multivariate polynomials integer partitions  Even Better  Starting point Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Probability to find exactly k real eigenvalues  The Answer a nonuniversal ingredient zonal polynomials Jack polynomials at α=2

  26. ] [  17 Applied Mathematics 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 !!

  27. ] [  16 GOE characteristic polynomial Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: I. Integrating out j‘ s cancellation  Reduced integral representation Nagao-Nishigaki (2001), Borodin-Strahov (2005)  Starting point

  28. ] [  15 not a projection operator ! Dyson Integration Theorem Inapplicable !! –part of a GOE matrix kernel Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: I. Integrating out j‘ s  Reduced integral representation GOEskew-orthogonal polynomials How do we calculate the integral ?..

  29. ] [  14 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom 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 ?

  30. ] [  13 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem Two fairly compact proofs

  31. ] [  12 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem Apply !!

  32. ] [  11 a probability to have all eigenvalues real Zonal polynomials a nonuniversal ingredient Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem Solved !!

  33. ] [  10 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem

  34. ] [  09 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem  Fredholm Pfaffian (Rains 2000)

  35. ] [  08 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem   

  36. ] [  07 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem

  37. ] [  06 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem

  38. ] [  05 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Sketch of derivation: II. Pfaffian integration theorem

  39. ] [  04 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Outline reminder  Ginibre’s random matrices • Definitions & physics applications Ginibre’srealrandom 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. ] [  03 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » Conclusions  Statistics of real eigenvaluesin GinOE  Exact formula for thedistribution of the number k of real eigenvalues in the spectrum of n × n random Gaussian real (asymmetric) matrix  Solution highlights a link betweenintegrable structure ofGinOEand the theory ofsymmetric functions  Even simpler solution is found for the entiregenerating functionof the distribution of k  Pfaffian Integration Theorem as an extension of theDyson Theorem(far beyond the present context)

  41. ] [  02 << 1 directed chaos GinOE model ~ 1 Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra » What is next ? ! ?  Looking for specific physical applications (weak non-Hermiticity)  Asymptotic analysis of thedistribution of k(matrix size n taken to infinity) work in progress  Asymptotic analysis of thedistribution of k(when k scales with E[k] and the matrix size n that is taken to infinity)  Furtherextensionof thePfaffian integration theoremto determineallpartial correlation functions

  42. ] [  01 Statistics ofReal Eigenvalues in GinOE Spectra Eugene Kanzieper Department of Applied Mathematics H.I.T. - Holon Institute of Technology Holon 58102, Israel Alexei Borodin (Caltech) Gernot Akemann (Brunel) Phys. Rev. Lett. 95, 230501 (2005) arXiv: math-ph/0703019 (J. Stat. Phys.) in preparation Applied Mathematics Statistics of Real Eigenvalues in GinOE Spectra Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007

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