# RandomMatrixTheory

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## The Unreasonable Utility of Random Matrix Theory

In 1959 Eugene Wigner delivered the Richard Courant Lecture entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences at New York University. Perhaps the best evidence for his argument concerning the interplay of mathematics and the applied sciences actually comes from random matrix theory (RMT), to which he himself made pioneering contributions. Historically RMT emerged from diverse sources - Adolf Hurwitz's 1897 study on invariant measures of the classical groups; Wishart's 1928 investigations of the singular values of random, real rectangular matrices; and Wigner's famous "Semi-circle Law" for the global density of eigenvalues of a random symmetric matrix.

And since his observations the remarkable synergy between abstract areas of pure mathematics and a bewildering array of diverse applications in contemporary studies of RMT has grown. Some of the modern examples include:

• wireless communications TV_2004;
• statistical inference on high-dimensional datasets Joh_2006;
• efficiency of numerical algorithms ESW_2014;
• number theory and conjectures on the Riemann zeta-function MS_2005;
• the stability of large ecosystems May_1972;
• interfacial growth processes SS_2010.

Yet another example is the role of integrable systems in RMT as an essential description of the averages, correlations and fluctuations arising in the theory. This was the theme of the keynote lecture that Professor Percy Deift of the Courant Institute NYU gave as the MSRI-Clay Distinguished Professor for the program Random matrices, interacting particle systems, and integrability'' at the MSRI in 2010. Some introductory articles on these aspects of RMT are Dei_2000, Dia_2005 and Cip_1999, whereas an encyclopediac resource is the monograph For_2010.

Deift's observation is even more true now. We now have additional problems arising in RMT, such as the recent work on the singular values of products of rectangular matrices with random, complex i.i.d normal variables WF_2016, Str_2014 (the so-called Ginibre ensembles), where higher order non-linear differential equations (fourth order in this case) are key to an understanding of the problem. This generalises a famous result of Tracy and Widom in 1994 TW_1994a treating the smallest squared-singular value of a single random matrix with unitary symmetry, the so-called hard edge problem. Here they identified the third Painlevé equation as the relevant integrable system. This identification is not "accidental" or "gratuitous" in any sense but is in fact an essential step which allows the employment of powerful tools from analysis for studying the averages, correlations, large deviations etc of the models.

The projects envisaged here will exploit this connection by employing the tools of RMT and integrable systems in related but novel applications and so will suit students with a background in probability theory and classical analysis.

Cip_1999 B. Cipra. A prime case of chaos, 1999.

Dei_2000 P. Deift. Integrable systems and combinatorial theory. Notices Amer. Math. Soc., 47(6):631--640, 2000.

Dia_2005 P. Diaconis. What is a random matrix? Notices Amer. Math. Soc., 52(11):1348--1349, 2005.

ESW_2014 A. Edelman, B.D. Sutton, and Y. Wang. Random matrix theory, numerical computation and applications. In Modern aspects of random matrix theory, volume~72 of Proc. Sympos. Appl. Math., pages 53--82. Amer. Math. Soc., Providence, RI, 2014.

For_2010 P.J. Forrester. Log Gases and Random Matrices, volume~34 of London Mathematical Society Monograph. Princeton University Press, Princeton NJ, first edition, 2010.

Joh_2006 I.M. Johnstone. High dimensional statistical inference and random matrices. In International Congress of Mathematicians. Vol. I, pg 307--333. Eur. Math. Soc., Zürich, 2006.

May_1972 R.M. May. Will a large complex system be stable? Nature, 238(5364):413--414, August 1972.

MS_2005 F. Mezzadri and N.C. Snaith, editors. Recent perspectives in random matrix theory and number theory, volume 322 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005.

SS_2010 T. Sasamoto and H. Spohn. One-dimensional Kardar-Parisi-Zhang equation: An exact solution and its universality. Phys. Rev. Lett., 104:230602, Jun 2010.

Str_2014 E. Strahov. Differential equations for singular values of products of Ginibre random matrices. J. Phys. A, 47(32):325203, 27, 2014.

TW_1994a C.A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289--309, 1994.

TV_2004 A.M. Tulino and S. Verdú. Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, 1(1):1--182, 2004.

WF_2016 N.S. Witte and P.J. Forrester. Singular values of products of Ginibre random matrices. 2016.

RMT occupies an interface between many subdisciplines of mathematics and is increasingly relevant in a diversity of applications. As of 20-1-2012 the ISI database has 5744 published articles with RM in the title or abstract and the yearly amounts are given in the Figure below.

### Combinatorics

Random permutations, longest increasing sub-sequences, non-intersecting random walkers, poly-nuclear growth models last passage percolation models, queuing models.

### Mathematical Statistics

Principal components analysis, sample covariance matrices, Wishart distribution, null and non-null SCM, Zonal and Jack polynomials.

### Analysis

Fredholm determinants, Painlevé equations, orthogonal and bi-orthogonal polynomials, Riemann-Hilbert problems.

### Algebraic Geometry

Matrix Integrals and genus expansions.

### Number Theory

Moments of the Riemann Zeta function, distribution of the non-trivial zeros.

### Applications

• Antenna Networks and Wireless Communication
• Ecological Systems, Biogeographic pattern of species nested-ness, ordered binary presence-absence matrices, distribution of mutation fitness effects across species, Fisher's geometrical model,
• Financial Modelling, cross correlations of financial data
• Entanglement of Quantum States
• Quantum Chaos, semi-classical approximation.
• Quantum transport in mesoscopic systems
• data analysis and statistical learning
• Stable signal recovery from incomplete and inaccurate measurements
• Compressed sensing, best {$k$}-term approximation, {$n$}-widths