Free Random Variables and Matrix Analog ue of the Classical Probability Theory .

1. Maciej A. Nowak Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University Kraków, Poland Free Random Variables and Matrix Analogue of the Classical Probability Theory. Potentials for Complexity Science forBusiness, Governments, and the Media Collegium Budapest, August 3-5, 2006 (Z.BURDA, E.GUDOWSKA-NOWAK, R.JANIK, A.JAROSZ, J.JURKIEWICZ, G.PAPP, I.ZAHED) In http://csrc.if.uj.edu.pl

2. Information stored in the form of huge matrices [X]it • Typical examples • Climate studies – time points t, observation station i • Financial/ economic data • Information retrieval and search engines • Wireless technology • Population models, pattern recognition • Genetic chips, adjacency matrices….. Standard approach: statistical approach for matrices – random matrix theory

3. Can we do better? Can we have more user-friendly version of RMT ? Can we still benefit from the intuitive power of classical probability calculus? Can we gain more at operational level? Can we achieve new insights in the field of matrix complexity? YES : FREE RANDOM VARIABLES [Voiculescu, 90’]

4. LAW OF LARGE NUMERS ? CENTRAL LIMIT THEOREMS ? STABLE DISTRIBUTIONS? EXTREME EVENTS STATISTICS?

5. x, p(x) Independence Characteristic function Generating moments Additive cumulants Addition law M, P(M) Freeness Green’s function FRV = Matrix generalization of the classical probability

6. FRV Calculus • Knowing rA(l) for matrix ensemble A and knowing rB(l) for matrix ensemble B we can get rA+B(l) (addition law) for matrix ensemble A+B • Similar multiplication law for A x B • Central limit theorems (“Gauss”=Wigner semicircle) • Spectral heavy tails (Lévy) • Exact calculus for infinite random matrices

7. “Real Life” Examples: • A: Business -- Estimators for financial correlation matrices (2001) • B: Governments – Cross-correlations in macro-economy times series (2005) • C: Media -- Capacity bounds and model design of wireless telecommunication systems (1999)

8. A: Covariance C ij =SXitXjt /T • EWMA S(Xit Ett’ Xjt’ )/T [Morgan-Stanley] • E = diag[T(1-a)a**(t-1)/(1-a**T)] • N/T=0[Kondor,Pafka,Potters;cond-mat/0402573] • N/T =r (arbitrary)[our FRV approach, physics/0603024] (free product of known random C and deteministic known E)

9. B: Cross-correlations G ij =SYitXjt[Bouchaud et al, physics/0512090] • X – N inputs , Y – M outputs, at time t • Singular values are square roots of GG^T • YX^TXY^T ~ (Y^TY)(X^TX) (free product of “cleaned” Wishart matrices for inputs and outputs) 76 macroeconomicindicators (production, retail sails, consumer/production indices, interest rates, WTI oil….) versus 34 indicators of inflation (Composite Price Indexes) for 265 months (06-1983 till 07-2005)

10. C: Wireless world [Tulino-Verdu, Found. And Trends in Comm. and Inf. Theory 1(2004) 1] • Tse (1999) – decoupling of effective interferences in CDMA systems is nothing but the additivity of free random variables • Exacts bounds for Shannon capacity • Ideal modeling tool for MIMO systems • Capacity(z) = <ln det (z+sinr H†H)> • Derivative of capacity with respect to z is Green’s function (note ln detA=Tr ln A)

11. Summary and Prospects FRV- new game in complexity ‘town’ Promising research in statu nascendi: • DYNAMICS: Brownian crowd evolution versus classical Brownian walker, non-linear”Fokker_Planck”, phase transitions – new insight for the dynamical propagation of matricial information. • DISSIPATION; Certain FRV tools extendable for non-symmetric matrices (lost information, causal effects (time lags), directed networks, asymmetries) • FLUCTUATIONS; Promising fundamental research for relations between higher-point Green’s functions and concepts of freeness • (see references in the talk in the proceedings for newest research on above three issues)

12. Definition of average • Spectral density

13. Definition of average • Spectral density

14. Brézin, Zee, 1996,1998

16. Additivity law

17. Nontrivial since… TIME EVOLUTION OF MATRIX-LIKE INCREMENTS ??

18. Evolving Random Matrices pdf for y(τ) fulfills diffusion (Fick’s equation) Matrix analog: free additive increments

19. Dyson, 1962; Voiculescu, 1993… Evolving Random Matrices Random walk…of large matrices

20. Evolving Random Matrices Random walk…of large matrices „Free” Fick/ Fokker-Planck Equation

21. Dyson 1962-stochastic diffusion of eigenvalues • For all t > 0, with Probability 1. • The process is given as a solution of the stochastic • differential equations, • where are independent standard one-dim. Brownian motions . • This process is called Dyson’s Brownian motion model. • Strong repulsive forces emerge among any pair of particles

22. Evolving Random Matrices Multiplicative Brownian diffusion EGN, Jurkiewicz, Janik, Nowak Matrix analogue: Spectrum of Y via S-transform analysis

23. EGN, Janik, Jurkiewicz, Nowak Multiplicative Matrix Brownian Motion Burger’s evolution equation!

24. Multiplicative Matrix Brownian Motion Snapshots from numerical simulations…

25. FRV- a powerfull shortcut in theory of probability • Random „plus” random (deterministic) - R transform • Random „times” random - S transform • Generalization for non-hermitian ensembles possible (Feinberg, Zee, Nucl. Phys. B, 504, 579; (1997); Janik, Nowak, Papp, Zahed Nucl. Phys. B, 501, 603, (1997) ) • Non-Gaussian measures (including Lévy) • Physical world: condensed matter (CPA), resonant states in nuclear reaction models, information theory (SINR maximalization), models for QCD, spectra of comples molecules… • Fundamentals ???: stationarity in diffusion, analogs of Boltzmann equilibrium, notion of entropy …

28. provided

30. Three Standard (Wigner-Dyson) Random Matrix Ensembles [1] The distribution of Eigenvalues of Hermitian Matrices in the Gaussian Unitary Ensemble (GUE) is given in the form [2] The distribution of Eigenvalues of Real Symmetric Matrices in the Gaussian Orthogonal Ensemble (GOE) is given in the form [3] The distribution of Eigenvalues of Quternion Self-Dual Hermitian Matrices in the Gaussian Symplectic Ensemble (GSE) is given in the form