Title: Sum rules via large deviations
Abstract: We show a large deviation principle for the weighted spectral
measure of random matrices corresponding to a general potential. Unlike
for the empirical eigenvalue distribution, the speed reduces to n and
the rate function contains a contribution of eigenvalues outside of the
limit support. As an application, this large deviation principle yields
a probabilistic proof of the celebrated Killip-Simon sum rule: a
remarkable relation between the entries of a Jacobi-operator and its
spectral measure. The talk is based on joint works with Fabrice Gamboa
and Alain Rouault.
Title: Planar orthogonal polynomials and boundary universality in the random normal matrix model.
Abstract:
This reports on recent joint work with Håkan Hedenmalm.
Motivated by questions concerning the boundary behavior of the
correlation kernel in the random normal matrix model, we study planar
orthogonal polynomials with respect to exponentially varying weights.
We obtain a complete asymptotic expansion of the orthogonal
polynomials, reminiscent of Carlemans classical theorem on planar
orthogonal polynomials on a simply connected domain, which allows us to
obtain the universal boundary decay profile of the eigenvalues.
In the talk we will discuss this asymptotic expansion, in
particular we focus on a new technique which decomposes planar
orthogonality into orthogonality along a curve family which foliates a
planar region.
The talk is a continuation of last week's analysis seminar given by Håkan, but we aim for it to be self-contained.
Title: RSK integrabilities
Abstract: In this talk I'll review two kinds of the
so-called integrabilities of the Robinson-Schensted-Knuth (RSK)
type dynamics, using the RSK and the qRSK algorithms as examples.
I'll also show that the two integrabilities are equivalent, at least in the Macdonald case,
if the corresponding RSK dynamics exists.
Title: To infinity and back (a bit)
Abstract: This is a colloquium style talk. Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some spectral information about H, such as the continuity of its spectral measure, what can be said about the asymptotic spectral properties of its finite dimensional approximations? This is a natural (and general) question, and can be used to frame many specific problems such as the asymptotics of zeros of orthogonal polynomials, or eigenvalues of random matrices. We shall discuss some old and new results in the context of this general framework and present various open problems.
Title: Determinantal structures in (2+1)-dimensional growth and decay models
Abstract: I will talk about an inhomogeneous growth and decay model with a wall present in which the growth and decay rates on a single horizontal slice of the surface can be chosen essentially arbitrarily depending on the position. This model turns out to have a determinantal structure and most remarkably for a certain, the fully packed, initial condition the correlation kernel can be calculated explicitly in terms of one dimensional orthogonal polynomials on the positive half line and their orthogonality measures.
Title: Periodic weights for lozenge tilings of hexagons
Abstract: The model of lozenge tilings of a hexagon is equivalent to a model of non-intersecting paths on a discrete lattice, i.e. the positions (or heights) of these paths, as well as their domains (times), are discrete. We will show how this correspondance works, supported by pictures. The discrete lattice can be viewed as a graph where we assign a weight on the arrows joining the vertices. In the simplest case, the weight function is periodic of period 1 in both space and time directions. We review some known results in this case. Then, we introduce some larger periodicities in the weight. We will focus on the case when the weight is periodic of period 2 in both space and time directions. The heights of the paths at a given time in a determinantal point process. In the second part of the talk, based on a recent preprint by M. Duits and A.B.J. Kuijlaars, we introduce matrix valued orthogonal polynomials of size 2x2 related to a 4x4 Riemann-Hilbert problem (RHP), and we show how the correlation kernel can be expressed in terms of this RHP. We will show some steps in the Deift/Zhou steepest descent which does not appear in usual steepest descent for 2x2 RHPs. We will finish by showing some pictures when we allow higher periodicities in the weight. This talk is intended to be accessible to non-specialists, and is based on a work in progress with M. Duits, A.B.J. Kuijlaars and J. Lenells.
Title: On the centered maximum of the Sine beta process
Abstract: There has been a great deal or recent work on the asypmtotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered.
Title: Dimer models
Abstract: In this talk I will show how to find the intrinsic conformal structure of the variational problem for dimer models. Moreover, I will introduce a natural class of domains and boundary values for the dimer models, and show how the conformal structure induces a strong rigidity of the liquid regions. In particular, this leads to a classification of the regularity of the liquid regions for all dimer models. Using this, I will discuss certain conjectures for the local boundary fluctuations. In a follow up talk on the analysis seminar April 25, I will discuss more of the analytical difficulties of the variational problem from a PDE perspective.
Title: Double integral formulas for periodic multi-level particle systems
Abstract: Recently, important progress has been made on the asymptotic behavior of certain periodic multi-level particle systems, such as the periodic weightings of domino tilings of the Aztec diamond and periodic weightings of lozenge tilings of a hexagon. In a general setting Duits and Kuijlaars recently proved a double integral formula for the kernel of the point process in terms of matrix valued orthogonal polynomials. I will discuss a simplification in the case of certain particle systems with infinite levels. In those cases, the integrand in the double integral formula can be expressed in terms of a matrix Wiener-Hopf factorization for an associated weight function. If there is time, I will mention the solution of the factorization problem for the $ 2\times 2 $ periodic weighting of lozenge tilings in a hexagon.
Title: Symmetric last passage percolation and Schur Processes
Abstract: We consider the last passage percolation (LPP) model on the square lattice with symmetric weights.
Tuning the weights on the diagonal and the endpoint, we obtain a crossover between the Tracy-Widom GUE, GOE and GSE distributions from random matrix theory. The LPP time can be seen as a marginal of a certain point process - the Schur process with free boundaries - and we shall explain this connection and how it can be used to obtain our results. Joint work with Jérémie Bouttier, Dan Betea and Mirjana Vuletic.
Title: Random tableaux and sorting networks
Abstract: Fix your favourite set of combinatorial objects, and
pick a large one at random. For various families of combinatorial
objects the result will be (after a suitable scaling, with probability
one) close to a unique limit object. I will talk about limit shapes for
tableaux and sorting networks.
Title: Level repulsion for arithmetic toral point scatterers
Abstract: The Seba billiard was introduced to study the
transition between integrability and chaos in quantum systems. The
model seem to exhibit intermediate level statistics with strong
repulsion between nearby eigenvalues (consistent with random matrix
theory predictions for spectra of chaotic systems), whereas large gaps
seem to have "Poisson tails" (as for spectra of integrable systems.) We
investigate the closely related "toral point scatterer"-model, i.e.,
the Laplacian perturbed by a delta-potential, on 3D tori of the form
R^3/Z^3. This gives a rank one perturbation of the original Laplacian,
and it is natural to split the spectrum/eigenspaces into two parts: the
"old" (unperturbed) one spanned by eigenfunctions vanishing at the
scatterer location, and the "new" part (spanned by Green's functions).
We show that there is strong repulsion between the new set of
eigenvalues.