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: Competition interface in the exactly-solvable exponential corner growth model with inhomogeneous parameters.
Abstract: The corner growth model with exponential weights is a much studied stochastic planar growth model in the Kardar-Parisi-Zhang universality class. The interest in the model stems in part from its exact-solvability and connections to fundamental interacting particle systems such as TASEP. In this talk, we consider an inhomogeneous generalization of the exponential CGM by allowing the means of the weights to be site-dependent. In this model, we study the competition interface (Ferrari and Pimentel, 2005), a notion of boundary that separates two growing planar regions that are in competition for sites. We observe that the inhomogeneity of the weights causes a sharp dichotomy in the asymptotics: A.s., either the competition interface has an asymptotic direction or it remains confined in a narrow strip of the quadrant with finite width. We discuss this result and its counterparts for the related notions of Busemann limits and second-class customers in a sequence of servers. This is an ongoing joint work with C. Janjigian and T. Seppalainen.
Title: Applications of the theory of Gaussian multiplicative chaos to random matrices.
Abstract: Log-correlated fields are a class of stochastic processes which describe the fluctuations of some key observables in different probabilistic models in dimension 1 and 2. For instance in percolation theory, random domino tilings or the characteristic polynomials of random matrices. Gaussian multiplicative chaos is a renormalization procedure which aims at defining the exponential of a Log-correlated field in the form of a family of random measures. These random measures can be thought of as describing the extreme values of the underlying field. In this talk, we shall present some applications of the theory to study the logarithm of the characteristic polynomial of some random matrices, this will include the Gaussian unitary ensemble, circular beta ensembles and the Ginibre ensemble.
Title: Random permutations and the Heisenberg model.
Abstract: We discuss probabilistic representations of certain quantum spin systems, including the ferromagnetic Heisenberg model, in terms of random permutations. The cycle structure of the random permutations is connected with the correlation structure in the spin-system, and it is expected that this cycle structure converges to a distribution known as Poisson--Dirichlet, in the limit of large systems. This problem is still open but we present some partial progress.
Title: On local geometry and spectrum of graphs
Abstract: The spectrum of a matrix is like its soul while the kernel is the body. An old question, of course, is how much of a body is remembered by the soul. Metaphors aside, I will speak about how the spectrum of big graphs determine their local geometry around typical vertices and the relationship between spectra and density of short cycles. For instance, certain graphs can be identified by the spectral radius alone. Along the way there will be connections to non-backtracking walks, entropy, graph limits and amenability.
Title: Localization and delocalization for ultrametric random matrices
Abstract: We consider a Dyson-hierarchical analogue of power-law random band matrices with Gaussian entries. This model can be constructed recursively by alternating between averaging independent copies of the matrix and running Dyson Brownian motion. We use this to map out the localized regime and a large part of the delocalized regime in terms of local statistics and eigenfunction decay. Our method extends to a part of the delocalized regime in which the model has a well-defined infinite-volume limit with Holder-continuous spectral measures. This talk is based on joint work with Simone Warzel.
Title: A (2+1)-dimensional Anisotropic KPZ growth model with a smooth phase
Abstract: Stochastic growth processes in dimension (2+1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes known as the isotropic KPZ class and the anisotropic KPZ class (AKPZ). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class, fluctuations are logarithmic in time and space. These classes are determined by the sign of the determinant of the Hessian of the speed of growth. It is natural to ask (a) if one can exhibit interesting growth models with "smooth" stationary states, i.e., with O(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf's picture) and (b) what new phenomena arise when the speed of growth is not smooth, so that its Hessian is not defined. These two questions are actually related and in this talk, we provide an answer to both, in a specific framework. This is joint work with Fabio Toninelli (CNRS and Lyon 1).
Title: Asymptotic trace formulas for random Schrodinger operator and
related problems of quantum informatics
Title: Last passage times in discontinuous inhomogeneous environments.
Abstract: We are studying a last passage percolation model on the two dimensional lattice, where the environment is a field of independent random exponential weights with different parameters. Each variable is associated with a lattice vertex and its parameter is selected according to a discretization of lower semi-continuous parameter function that may admit discontinuities on a set of curves. We prove a law of large numbers for the sequence of last passage times, defined as the maximum sum of weights which a directed path can collect from (0, 0) to a target point (Nx, Ny) as N tends to infinity and the mesh of the discretisation of the parameter function tends to 0 as 1/N. The LLN is cast in the form of a variational formula, optimised over a given set of macroscopic paths. Properties of maximizers to the variational formula above are investigated in two models where the parameter function allows for analytical tractability. This is joint work with Federico Ciech.
We are currently not hiring new postdocs, but in the fall of 2019 a new announcement will come out for one or more postdoctoral positions.