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.
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.