In many situations we have a stochastic microscopic model of some kind and we want to understand what happens for large such systems. Often a macroscopic, non-random shape of some form emerges. This limit shape typically depends on the details of the model and is often described by some variational principle or a differential equation. If we look at the microscopic details of the model they will also depend on the specific model. However, random fluctuations of the the limit shape at an intermediate scale are very often universal, i.e. they are the same within broad classes of models. This phenomenon is called universality. A classical example is the central limit theorem which is a universal limit law in many situations. It is a very interesting and surprising fact, that has only emerged in the last 15-20 years, that in many cases the fluctuations around the limit shape are the same as the distributions that appear in random matrix theory and these laws appear to be universal.One basic example is the Tracy-Widom distribution for the largest eigenvalues of big matrices. Investigating this phenomenon and proving the occurrence of these laws in various models has been a central research theme for the group at KTH. Typical models are local random growth models, directed polymers and random tiling/dimer models.
The figure on the right shows a random uniform tiling of a shape called the Aztec diamond. We can see a shape emerging in this picture, namely the interface between the disordered region in the center and the completely ordered tiling surrounding it. This is interface has fluctuations described by the Tracy-Widom distribution which originated in random matrix theory. The mathematics behind these developments is very rich with connections to asymptotic analysis, combinatorics, representation theory, special functions, mathematical physics, probability theory and more. Some of the results have also been verified experimentally in random growth experiments. The area is very active at the moment with many new developments. Ongoing research at KTH is concerned for example with with models generalizing the Aztec diamond and deeper properties of random growth models.
Random matrices are an attractive class of models for large stochastic systems. Their study connects mathematics with many branches of science such as statistics, physics, computer science and genomics to mention a few. In statistics, random matrices were introduced by Wishart in 1928 as part of the statistical analysis of large samples as associated sample covariance matrices. In physics, they were introduced by Wigner in 1955 in the study of energy levels of heavy nuclei. Wigner envisioned that the energy levels of large complex quantum systems behave in the same way as the eigenvalues of large matrices whose entries are independent random variables, and that the emerging behavior is universal in the sense that it depends only on the basic symmetry type of the systems but is otherwise independent of the details. While we still do not understand universality for most physically realistic models, spectacular progress was made in the last decade in deriving universality for Wigner random matrices with parallel results for sample covariance matrices. The developed methods combine powerful tools from probability theory, functional analysis and mathematical physics. Interesting applications yielding a strong link to high dimensional statistical inference and stochastic growth models include the derivation of Tracy–Widom fluctuations of the largest eigenvalue for various random matrix ensembles. Ongoing research at KTH focuses on the understanding of the universality phenomena in more general and more physical models, on connections to quantum physics, spectral theory, integrable probability and the theory of free probability, as well as on applications in mathematical statistics and in the theory of random networks.
The research activities in our group are strongly related to analysis. In studying various universality questions of interest one often relies on methods and techniques from fields such as complex analysis, potential theory, ordinary differential equations and spectral theory. An important connection is with the theory of orthogonal polynomials and their asymptotic properties as the degree becomes large. In the late 90's the Riemann-Hilbert approach for those polynomials was introduced to obtain various asymptotic results, including a first proof of the universality for the microscopic behavior of the eigenvalues of Unitary Ensembles. This complex analytical tool proved to be very fruitful. In particular, in the study of singular behaviors or phase transitions where remarkable connections have been found with special functions such as the Painlevé transcendents. It has also been successfully applied to the asymptotic analysis of Toeplitz determinants with Fisher-Hartwig symbols. This is an active area with interesting ongoing developments.
A notion strongly related to orthogonal polynomials that plays a prominent role in random matrix theory are Jacobi matrices and CMV matrices. For instance, gaussian beta ensembles can be represented as eigenvalues of random Jacobi matrices or CMV matrices. This opens up interesting connections to spectral theory. A recent development that has been of particular interest in the research group at KTH, is the use of Jacobi matrices in the analysis of fluctuations of macroscopic and mesoscopic linear statistics for determinantal point processes. This is a new approach for understanding the global fluctuations and the random Gaussian fields that arise. A topic that currently witnesses a high activity in the literature.
Some references for previous research at KTH:
For all mathematics seminars and talks see the Stockholm Mathematics Kalendarium.
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.
We are currently not hiring new postdocs, but in the fall of 2018 a new announcement will come out for one or more postdoctoral positions.