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: *

- J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178
- Z. Bao, L. Erdos and K. Schnelli, Local law of addition of random matrices on optimal scale, Comm. Math. Phys. 349 (2017), no. 3, 947-990
- J. Breuer and M. Duits, Central Limit Theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, J. Amer. Math. Soc. 30 (2017) no. 1, 27-66. (arXiv:1309.6224)
- S. Chhita and K. Johansson, Domino statistics of the two-periodic Aztec diamond. Adv. Math. 294 (2016), 37–149
- M. Duits and D. Geudens, A critical phenomenon in the two-matrix model in the quartic/quadratic case, Duke Math. J. 162 (2013) no. 8, 1383--1462. (arXiv:1111.2162)
- M. Duits and A. Kuijlaars, The two periodic Aztec diamond and matrix valued orthogonal polynomials, arXiv (2017)
- M. Duits, On global fluctuations for non-colliding processes, Ann. Probab. 46.3 1279-1350 (2018)
- K. Johansson, Shape fluctuations and random matrices. Comm. Math. Phys. 209, no. 2, 437–476 (2000)
- K. Johansson, Two Time Distribution in Brownian Directed Percolation, Comm. Math. Phys. 351, no. 2, 441-492 (2017)
- R. Kozhan and M. Duits, Relative Szego Asymptotics for Toeplitz determinants, International Mathematics Research Notices (2016)
- J. O. Lee and K. Schnelli, Local law and Tracy-Widom limit for sparse random matrices, Probab. Theory Related Fields 171(1), 543-616 (2018)
- J. O. Lee and K. Schnelli, Tracy-Widom distribution for the largest eigenvalue of real sample covariance matrices with general population, Ann. Appl. Probab. 26, no. 6, 3786-3839 (2016)

- D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413–432
- A. Borodin and V. Gorin, Lectures on integrable probability, arXiv:1212.3351
- P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, vol. 3, New York, American Mathematical Society 2000
- I. Corwin, Kardar-Parisi-Zhang universality. Notices Amer. Math. Soc. 63 (2016), no. 3, 230–239
- L. Erdos and H.-T. Yau, Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc. 49(3), 377-414, (arXiv:1106.4986)
- K. Johansson, Kurt Random matrices and determinantal processes. Mathematical statistical physics, 1–55, Elsevier B. V., Amsterdam, 2006
- I. M. Johnstone, High Dimensional Statistical Inference and Random Matrices, Proc. International Congress of Mathematicians 2006, 307-333, (arXiv:math/0611589)
- A.B.J. Kuijlaars, Universality, in: “Oxford Handbook on Random Matrix theory”, edited by G. Akemann, J. Baik and P. Di Francesco, Oxford University Press, 2011.

For all mathematics seminars and talks see the Stockholm Mathematics Kalendarium.

**Title:** Local laws for polynomials of Wigner matrices.

**Abstract:** We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. First, we give a brief introduction to the linearization technique that allows to transform the polynomial model into a linear one, which has simpler correlation structure but higher dimension. After that we show that the local law holds up to the optimal scale for the generalized resolvent of the linearized model, which also yields the local law for polynomials. Finally, we show how the above results can be applied to prove the optimal bulk local law for two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries.

This is a joint work with Laszlo Erdos and Torben Kruger.

**Title:** Rigidity of eigenvalues for beta ensemble in multi-cut regime

**Abstract:** I will first talk about the background and some well known results of beta ensemble. Then I will introduce the rigidity of eigenvalues for beta ensemble in multi-cut regime, i.e., the fact that each eigenvalue in the bulk is very close to its "classical location". The probability that the distance between the eigenvalue and its classical location is larger than N^{-1+r} is exponentially small where r is an arbitrarily small positive number. The model is an generalization of the beta ensemble in one-cut regime for which the rigidity of eigenvalues was proved by Bourgade, Erdos and Yau. Finally I will explain the difference between the proof in one-cut case and the proof in multi-cut case.

**Title:** On shocks in the TASEP

**Abstract:** The TASEP particle system runs into traffic jams when the initial particle density increases in the direction of flow. It servers as a microscopic model of shocks in Burgers' equation. I will describe work with Jeremy Quastel on a specialization of TASEP, where we identify limiting joint fluctuations of particles at the shock by using determinantal formulae for its correlation functions. The limit is described in terms of random matrix laws.

**Title:** Optimal Clustering Algorithms in Block Markov Chains

**Abstract:** In this talk, we consider cluster detection in Block Markov Chains. These Markov chains are characterized by a block structure in their transition matrix. More precisely, the $n$ possible states are divided into a finite number of $K$ groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper we devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the Block Markov Chain and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense.

Joint work with Jaron Sanders and Seyoung Yun

**Title:** Universal Local Statistics of Lyapunov exponents

**Abstract:** We consider the product of M complex Ginibre random matrices of sizes
N × N, being a simple toy model for chaotic dynamical systems.
While the behaviour of the Lyapunov exponents of the product matrix was
known in the limit M to inifinity at finite N, taking deterministic
values,
recent progress has been made at finite M and N. The corresponding
exponents follow a determinantal point process and thus all correlation
functions of the squared singular values of the product matrix are
known. This has allowed us to take a double scaling limit for N,M to infinity,
where the Lyapunov exponents exhibits a rich variety of correlations,
interpolating between deterministic behaviour and the sine- or
Airy-kernel of a single random matrix, depending on the location in the
spectrum. Surprisingly, in the bulk we find the same limiting
interpolating kernel as for Dyson's Brownian motion for certain initial
conditions as constructed by Johansson.
This is joint work with Zdzislaw Burda and Mario Kieburg
[arXiv:1809.05905 [math-ph]].

**Title: **Level crossings in deterministic and random matrix pencils.

**Abstract:** Collisions of energy levels (alias level crossings

) in pencils of linear operators is a physically important phenomenon studied since the down of quantum mechanics. In this talk we discuss two intriguing concrete examples coming from the so-called quasi-exactly solvable potentials and mainly concentrate on some recent results about the distribution of level crossings in families A+t B, where A and B and independently taken from some standard random matrix ensembles.

**Title: **Gaussian Approximation of the Distribution of Strongly Repelling Particles on the Unit Circle

**Abstract:** Abstract: In this talk, we will consider an interacting particle system on the unit circle with stronger repelling than that of Circular beta ensemble. I will prove the Gaussian approximation of the distribution of the particles in this model. This is joint work with Alexander Soshnikov.

**Title:** SLEs and related conformally invariant processes

**Abstract:** I will give an informal introduction to the Schramm-Loewner evolution and (depending on time) connections to related topics such as Gaussian fields, Brownian loops and conformal field theory.

**OBS: Special Time: 13:00, F11**

**Title:** Random walks on groups and free convolutions.

**Abstract:** We start with a result of Harry Kesten stating that a symmetric random walk on the free group of order d is transient for d>1. I will sketch a proof that leads us to the concept of free independence of non-commutative random variables as introduced by Dan Voiculescu. A main tool in the proof is the free additive convolution of probability measures, an operation associated with the addition of freely independent random variables. Kesten's result follows from a special case of the free convolution which is accessible via explicit calculations. I will then discuss properties of the free convolution of generic probability measures and present some recent regularity results. I will conclude by establishing a link with random matrix theory and explain the role of free probability in some on-going research projects.

Spring 2018

We are currently offering two postdoctoral positions. One with Maurice Duits and one with Kevin Schnelli as mentors. The positions are for two years and start on July 1, 2019 or later. Deadline for applications is January 9, 2019. For more information click here.

Please note that applications have to be submitted via KTH's application system.