Abstract: Quantum graphs have been introduced by Kottos and Smilansky as particularly useful models in quantum chaos. These models are based on the spectral problem of a Laplacian on a compact metric graph. We extend these models by considering a Dirac operator instead of a Laplacian, thus allowing for symplectic symmetries. In this talk we present the construction of the models, including self-adjoint realisations of the Dirac operator that respect time-reversal invariance, as well as numerical studies of eigenvalue correlations.

Abstract: We present an elementary and self-contained derivation of formulae for products and ratios of characterisitc polynomials from classical groups using classical results due to Weyl and Littlewood (joint work with Dan Bump). Some combinatorial and number-theoretic applications will also be discussed.

Abstract: We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.

Abstract: I will discuss some recent numerical results relating to the percolation model for nodal domain statistics in quantum chaotic systems introduced by Bogomolny and Schmit.

Abstract: For many classically chaotic systems, it is believed that in the semiclassical limit, the matrix elements of smooth observables approach the phase space average of the observable. In the approach to the limit the matrix elements can fluctuate around this average. I will talk about these fluctuations, for the quantum cat map on the 4-dimensional torus. In particular, on maps and observables for which the variance has a different rate of decay than is expected for generic chaotic systems.

Abstract: Let A be a fixed nxn matrix and U be a unitary matrix picked up at random from the unitary group U(n). We express the integer moments of the spectral determinant |det(z-AU)|^2 in terms of the characteristic Quantum Leakspolynomial of the matrix AA^*. This result provides a useful tool for studing the eigenvalue distributions of complex random matrices. Links between this problem, Kaneko's generalization of the Selberg integral and Zirnbauer's color-flavor transformation will be discussed. [Joint work with Y.V. Fyodorov]

Abstract: We consider k i.i.d. random walks on the real line. Our goal is to construct the conditional version given that all walkers stay in a strict order at all times. This is the discrete analogue of Dyson's Brownian motions (also called non-collding Brownian motions), which is a system of k i.i.d. standard Brownian motions conditioned on having no collision of the particles at any time. The construction is via a Doob h-transform with h equal to the Vandermonde determinant.
Non-colliding versions of nearest-neighbour random walks on the integers have been constructed for a few explicit examples. The construction is also in terms of a Doob h-transform using a positive regular function h, which turns out to be the Vandermonde determinant in these particular cases.
In the talk I consider the case of general random walks (modulo some moment condition) and construct the non-colliding version via a Doob h- transform. The positive regular function h is strongly related to the Vandermonde determinant. We also give precise large-n asymptotics for the probability of being strictly ordered up to time n, including an invariance principle towards Dyson's Brownian motions.

Abstract: In my talk I will discuss recent results on the universality of local eigenvalue statistics for orthogonal and symplectic random matrix ensembles of Hermite and Laguerre type. The proof of these results is based on formulae introduced by Tracy and Widom which express the relevant quantities in terms of orthogonal polynomials (rather than in terms of skew orthogonal polynomials). The orthogonal polynomials in turn are analyzed using an associated Riemann-Hilbert problem.

Abstract: Cumulants provide a combinatorial way to understand stochastic independence. In noncommutative probability, various notions of the latter have been introduced and appropriate cumulants were found. We give a unified theory of these cumulants based on exchangeability. In classical probability, exchangeable sequences of random variables are characterized as conditionally i.i.d. by De Finetti's theorem. In the noncommutative situation, there is no general result like this. However we show that if the cumulants happen to be noncrossing and a certain ``weak singleton condition'' holds, then there must be a free amalgamated product involved.

Abstract: We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity---provided there is no extreme level clustering---and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.

Abstract: One studies the eigenvalue counting function for the operator $H = -\Delta + V$ with a continuous real-valued function $V$, on the two-dimensional torus $\mathbb T$. The Berry-Tabor hypothesis claims that large eigenvalues of quantum Hamiltonians with a completely integrable underlying classical system, must be distributed according to the Poisson law. In particular, this hypothesis applies to the operator $H_0 = -\Delta$ on $\mathbb T$. However, at present only partial results towards the proof of this conjecture are available. The aim of my talk is to show that the distributions of discrete spectra of $H$ and $H_0$ coincide irrespectively of the distribution law for the spectrum of $H_0$. More precisely, we prove that all moments of the counting function distributions for $H$ and $H_0$ asymptotically coincide in the large energy limit.

Abstract: We shall consider large Wigner and sample covariance random matrices in the case when matrix entries have heavy tail distributions. The main result states that, under some additional conditions, the largest eigenvalues of such matrices obey Poisson statistics in the limit of large dimension.

Abstract: The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fix time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.

Abstract: One of the source of random matrices is the theory of random structures which can be described with matrices (tensors). The examples are random graphs, metric spaces and so on. The main probelm appeared is the classification problem. It reduces to the description of the invrinat distributions on the spaces of matrices with some symmetries. For example.symmetry with respect to the group of permutations or orthogonal, unitary groups and so on. Our result concerns to the necessary and sufficient conditions on the random matrix which can be so called "matrix distribution" of the functions of two variables.

Abstract: We study the distribution of the number of lattice points lying in thin elliptical annuli. It has been conjectured by Bleher and Lebowitz that, if the width of the annuli tend to zero while their area tends to infinity, then the distribution of this number, normalized to have zero mean and unit variance, is Gaussian. This has been proved by Hughes and Rudnick for circular annuli whose width shrink to zero sufficiently slowly. We prove this conjecture for ellipses whose aspect ratio is transcendental and strongly Diophantine, also assuming the width shrinks slowly to zero.

Abstract: Building upon Dyson's fundamental 1962 article known in random matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Eliminating the unitary symmetries by transferring to an irreducible block of equivariant homomorphism, we show that each set of irreducible block data determines an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.