Eigenvalue statistics for beta-ensembles

Random matrix theory is a maturing discipline with decades of research in multiple fields now beginning to converge. Experience has shown that many exact formulas are available for certain matrices with real, complex, or quaternion entries. In random matrix jargon, these are the cases β = 1, 2 and 4 respectively. This thesis explores the general P > 0 case mathematically and with symbolic software. We focus on generalizations of the Hermite distributions originating in physics (the "Gaussian" ensembles) and the Laguerre distributions of statistics (the "Wishart" matrices). One of our main contributions is the construction of tridiagonal matrix models for the general (β > 0) 0 β-Hermite and (β > 0, a > β(m - 1)/2) β-Laguerre ensembles of parameter a and size m, and investigate applications of these new ensembles, particularly in the areas of eigenvalue statistics. The new models are symmetric tridiagonal, and with entries from real distributions, regardless of the value of β. The entry distributions are either normal or X, so "classical", and the independence pattern is maximal, in the sense that the only constraints arise from the symmetric/semi-definite condition. The β-ensemble distributions have been studied for the particular 1, 2, 4 values of p as joint eigenvalue densities for full random matrix ensembles (Gaussian, or Hermite, and Wishart, or Laguerre) with real, complex, and quaternion entries (for references, see [66] and [70]). In addition, general -ensembles were considered and studied as theoretical distributions ([8, 51, 50, 55, 56]), with applications in lattice gas theory and statistical mechanics (the parameter being interpreted as an arbitrary inverse temperature of a Coulomb gas with logarithmic potential).

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2003

Author dc:creator

• Dumitriu, Ioana, 1976-

Advisor dc:contributor.advisor

• Alan Edelman.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/29347

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/29347