Moderate, large, and superlarge Deviations for extremal Eigenvalues of unitarily invariant Ensembles

by Katharina Schüler

Institution: Universität Bayreuth
Department: Fakultätsübergreifende Einrichtung
Degree: PhD
Year: 2015
Record ID: 1101352
Full text PDF: https://epub.uni-bayreuth.de/2029/


A celebrated result in Random Matrix Theory is that the distribution of the largest eigenvalue of the Gaussian Unitary Ensemble converges (after appropriate rescaling) to the Tracy-Widom distribution if the matrix dimension N tends to infinity. The interest in this distribution rose even more when it turned out that it appears not only in the description of extremal eigenvalues for a large class of matrix ensembles but also provides the limit law for a variety of stochastic quantities in statistical mechanics. This phenomenon is called universality in Random Matrix Theory. It should be noted that the Tracy-Widom Law describes the distribution of the largest eigenvalue only in a neighborhood of its mean that has a size of order N^(-2/3). As the main result of this thesis we provide a complete leading order description with uniform error bounds for the upper tail of the distribution of the largest eigenvalue beyond the Tracy-Widom regime. In addition, we are not only concerned with the Gaussian Unitary Ensemble. Our results apply to unitarily invariant ensembles whose probability measure is parameterized by potentials in the class of real analytic and strictly convex functions. According to standard notation in stochastics, we study the upper tail in the regimes of moderate, large, and superlarge deviations. Our results are new except for a small region in the regime of moderate deviations of size (log (N)/N)^(2/3) that were proved by Choup and by Deift et al. They allow in particular to identify precisely the range of universality of the distribution of the largest eigenvalue. Moreover, we strengthen previous large deviations results of Anderson et al., Johansson, and Ledoux et al. In order to obtain our results on the distribution of the largest eigenvalue, we use the Orthogonal Polynomial method for unitarily invariant ensembles. The asymptotic analysis of the relevant Orthogonal Polynomials is then performed by the Riemann-Hilbert approach introduced by Deift et al. On a technical level our results are based on a new leading order description of the Christoffel-Darboux kernel in the region of exponential decay. Hereby we show in particular how the rate function, known from the theory of large deviations, is related to the Airy kernel that is usually used for the description in the Tracy-Widom regime as well as in the moderate regime. Some of our main results have been announced in joint work with Thomas Kriecherbauer, Kristina Schubert, and Martin Venker. In that paper a number of results of this thesis has been used in a slightly more general context. In der Theorie der Zufallsmatrizen war es eine bahnbrechende Erkenntnis, dass die Verteilung des größten Eigenwerts des Gaußschen unitären Ensembles (nach geeigneter Skalierung) gegen die Tracy-Widom-Verteilung konvergiert, sofern die Matrixdimension N gegen unendlich strebt. Diese Verteilung rief ein noch größeres Interesse hervor, nachdem sich herausgestellt hatte, dass sie sich nicht nur in der Beschreibung extremaler Eigenwerte einer großen Klasse von…