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Mathematics Institute

Thursday programme

Room B3.03

Morning session chair: Arno Kuijlaars

Time Speaker Title Abstract
09:00-09:50 Eugene Kanzieper Power spectrum analysis of long eigenlevel sequences in quantum chaology Fluctuations in quantum spectra are known to exhibit a high degree of universality which reflects the nature -- regular or chaotic -- of the underlying classical dynamics. Following Berry and Tabor (1977), statistics of level spacings in generic quantum systems with completely integrable classical dynamics is expected to mimic statistics of waiting times in a Poisson point process. For generic quantum systems with completely chaotic classical dynamics, Bohigas, Giannoni and Schmit (1984) conjectured that the level spacing distribution coincides with predictions of the Random Matrix Theory.

Recently, an alternative characterization of eigenvalue fluctuations was suggested by Relan\~{o} et. al. (2002). Interpreting long eigenlevel sequences as discrete-time random processes, these authors argued that the power spectrum of energy level fluctuations exhibits the $1/\omega$ behavior for completely chaotic and $1/\omega^2 $ behavior for completely regular quantum systems.

In this talk, we present a rigorous theory of the power-spectrum and show that it can be expressed in terms of Painlev\'e VI function. We also outline the asymptotic (large-$N$), analysis of the resulting expression to confirm the small--$\omega$ behavior reported in various numerical experiments. Further work is required to analyze behavior of the power-spectrum in the domain $\omega \sim O(N)$ and close to the Nyquist frequency.

This is a joint work with Vladimir Osipov (Lund) and Roman Riser (Holon).
09:50-10:40 Francesco Mezzadri Large deviations of radial statistics in the two-dimensional one-component plasma The two-dimensional one-component plasma is a ubiquitous model for several vortex systems. For special values of the coupling constant $\beta q^2$ (where q is the particles charge and β the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number $N$ of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading orders in $1/N$. We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine $1/N$-expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-$N$ formulae valid for $\beta q^2 = 2$. This is work in collaboration with Fabio Deelan Cunden and Pierpaolo Vivo
10:40-11:00 Coffee in the Common Room
11:00-11:50 Uzy Smilansky Spectral statistics of the Unimodular Matrix Ensemble - A trace formula approach. The Unimodular Ensemble (UME) consists of complex Hermitian $N\times N$ matrices with unimodular entries, with random phases which are uniformly distributed on the $N(N-1)/2$ dimensional torus. Together with Hans Weidenmueller ,we study some spectral statistics of the UME for large $N$ and compute the leading and the first non-vanishing correction in a $1/N$ expansion. Our purpose is to compare the trace-formula and the SUSY methods in addressing the same problem. In the present talk I shall mainly discuss the trace-formula approach.
11:50-12:40 Mario Kieburg

An intriguing Relation between Eigenvalues & Singular Values of
Bi-unitarily Invariant Random Matrices

 The relation between the eigenvalues and the singular values of an arbitrary complex square matrix is an old problem which was already studied by Schur, Weyl and others since the beginning of the last century. The only relations between the eigenvalues and singular values are given via inequalities for an arbitrary fixed matrix. This drastically changes when considering random matrices since we have an additional information due to the probability distribution. Very recently Holger Koesters and I could prove that there is a bijection mapping the joint probability distribution of the singular values to the one of the eigenvalues and vice versa for bi-unitarily invariant random matrix ensembles. I will sketch this proof in my talk and will outline the implications for spectral statistics.
12:40-14:00 Lunch in the Common Room