Random Matrix Theory (Math 8823 BED)

Schedule:

Monday  3:05-4:25PM  Skiles 279
Tuesday 4:05-5:25PM  Skiles 279

Content

Homework:

Series 1:  Laguerre polynomials
 
 

Final Exam (2002) : (ps) (pdf)
                                                              the report be given before Wednesday December 11th, 2002 at 5:00PM
                                                              in J. Bellissard's office (Skiles 132).

Three subjects are proposed below. One among them will be assigned to
each candidate. Each candidate will be asked to provide a written report
(handwritten, printed or {\LaTeX} file) on the assigned subject.

The report should contain:

(i) a presentation of the topic,
(ii) a short history,
(iii) a comprehensive bibliography,
(iv) the list of the main results,
(v) at least one example of calculation,
(vi)  the proof of at least one important mathematical result,
(vii) at least one important application to physics together with the
description of the corresponding (theoretical and experimental) results,
(viii)  a conclusion.

The   length of this report should not be more than 10  LaTeX or 20 handwritten pages.

The evaluation of the work will take into account both the content and the quality of the text and the presentation.
A guideline is proposed for each topic. The candidates can find a short list of  references to start with below.

Topic #1: The Gaussian Orthogonal Ensemble
Topic #2: The Voiculescu free entropy
Topic #3: Supersymmetric Methods
 
 

Books:

C.E. Porter (Ed.), Statistical theories of spectra: fluctuations, Academic Press, New-York (1965).
M. Mehta, Random Matrices, 2nd Ed., Acad. Press, (1990),
. V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, American Mathematical Society, (1992)
K. B. Efetov,  Supersymmetry in Disorder and Chaos,  Cambridge University Press, (1997).
F. Hiai, D. Petz,  The Semicircle Law, Free Random Variables and Entropy,
American Mathematical Society, (2000)
P.Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach ,
Courant Lecture  Notes, 3, (2000)
R. Speicher, Combinatorics of free probability theory,  Lecture Notes, I.H.P. Paris, (1999).

Articles:

J. Wishart, The generalized product moment distribution in samples from a normal multivariate population, Biometrika, 20A, (1928), 32-52.
F.J. Wegner, Disordered system with n orbitals per site:  n=[infty]   limit, Phys. Rev. B, 19, (1979), 783-792.
L. Schafer, F.J. Wegner, Disordered system with n orbitals per site: Lagrange Formulation, Hyperbolic Symmetry, and Goldstone Modes , Z. Phyzik B, 38, (1980), 113-126.
K. B. Efetov,  Supersymmetry and theory of disordered metals, Advances in Physics, 32, (1983), 53-127.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability I., Commun. Math. Phys., 155, (1993), 71-92.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability II., Invent. Math., 118, (1994), 411-440.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability III., Geom. Funct. Anal., 6, (1996),  172-199.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability IV., in Free Probability theory, D. V. Voiculescu (Ed.), Field Inst. Commun. 12, AMS, (1997), pp. 293-302.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability V., Invent. Math., 132, (1998), 189-227.
D.V. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability VI.,  Adv. Math., 146, (1999), 101-166.
P. Neu , R. Speicher, Rigorous mean-field theory for coherent potential approximation: Anderson model with free random variables, J. Stat. Phys., 80, (1995), 1279-1308.
P. Neu , R. Speicher, Random matrix theory for CPA: Generalization of Wegner's $n$--orbital model, J.  Phys., A28, (1995), L79-L83.
R. Speicher, Free Probability Theory and Non-Crossing Partitions , Séminaire Lotharingien de Combinatoire, B39c (1997), 38pp.
E. Brézin, Dyson's universality in generalized ensembles of random matrices, in The Mathematical Beauty of Physics, J.M. Drouffe & J.B. Zuber (eds.), World Scientific, (1997), pp.1-11.
P.W. Brouwer, On the Random-Matrix Theory of Quantum Transport, Ph.D. ThesisLeiden, June 1997.
T. Guhr, A, Müller-Groeling, H.A. Weidenmüller, Random matrix theory in quantum physics,, Phys. Rep., 299, (1998), 190-425.
U. Haagerup, S. Thorbjørnsen, Random Matrices with Complex Gaussian Entries,  preprint (1998).
U. Haagerup, S. Thorbjørnsen, Random Matrices and K-theory for Exact C*-algebras,  Documenta Mathematica, 4, (1999), 341-450.
G. Ben Arous, O. Zaitouni,  Large deviations from the circular law, ESAIM: Probability and Statistics, 2 (1998), 123-134.
P.Deift, T. Kriecherbauer, K. T.-R. MacLaughlin, 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, (1999), 1335-1425.
P.Deift, T. Kriecherbauer, K. T.-R. MacLaughlin, S. Venakides, X. Zhou, Uniform Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights,  Commun. Pure Appl. Math., 52, (1999), 1491-1452.
  L. Laloux, P. Cizeau, J.-Ph. Bouchaud, M. Potters, Noise dressing of financial correlation matrices , Phys. Rev. Letters, 83, (1999), 1467
A. Mirlin, Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach, Phys. Rep.,  326, (2000), 259-382.