The Polynomial Method for Random Matrices
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We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases.
Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form.
In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.
KeywordsRandom matrices Stochastic eigenanalysis Free probability Algebraic functions Resultants D-finite series
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- 3.G. Anderson, O. Zeitouni, A law of large numbers for finite-range dependent random matrices, Preprint, September 2006. http://arxiv.org/abs/math/0609364.
- 6.P. Biane, Free probability for probabilists, in Quantum Probability Communications, vol. XI, Grenoble, 1998 (World Scientific, River Edge, 2003), pp. 55–71. Google Scholar
- 10.P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Commun. Pure Appl. Math. 52, 1335–1425 (1999). MATHCrossRefMathSciNetGoogle Scholar
- 11.P.A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3 (New York University Courant Institute of Mathematical Sciences, New York, 1999). Google Scholar
- 12.W.B. Dozier, J.W. Silverstein, On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices (2004). http://www4.ncsu.edu/~jack/infnoise.pdf.
- 14.P. Flajolet, R. Sedgewick, Analytic combinatorics: functional equations, rational and algebraic functions, Research Report 4103, INRIA (2001). http://algo.inria.fr/flajolet/Publications/FlSe01.pdf.
- 15.F. Hiai, D. Pet, The Semicircle Law, Free Random Variables and Entropy, vol. 77 (American Mathematical Society, Providence, 2000). Google Scholar
- 20.R.R. Nadakuditi, Applied Stochastic Eigen-Analysis, PhD thesis, Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology, February 2007. Google Scholar
- 22.N.R. Rao, RMTool: a random matrix and free probability calculator in MATLAB. http://www.mit.edu/~raj/rmtool/.
- 23.N.R. Rao, A. Edelman, The polynomial method for the eigenvectors of random matrices, Preprint. Google Scholar
- 30.R.P. Stanley, Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999). With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. Google Scholar
- 31.B. Sturmfels, Introduction to resultants, in Applications of Computational Algebraic Geometry, San Diego, CA, 1997. Proceedings of Symposia in Applied Mathematics, vol. 53 (American Mathematical Society, Providence, 1998), pp. 25–39. Google Scholar
- 32.A.M. Tulino, S. Verdú, Random matrices and wireless communications, Found. Trends Commun. Inf. Theory 1 (2004). Google Scholar