# Mathematical physics and probability reading seminar

*(Covering topics related to random matrices, representation theory, integrable systems and interacting stochastic particle systems)*

**Seminars are held on Tuesdays at 12:00, B3.02**

## Term 1

- 30/09/2014 No seminar
**07/10/2014**Oleg Zaboronski.I will review two classes of asymptotically exact integrals arising in random matrix theory - the celebrated Itzykson-Zuber integral and its close counterpart appearing in the theory of the real Ginibre ensemble and non-unitary invariant GUE and GOE ensembles. In particular, I will show how the heat equation on Lie algebra su(N) can be used to calculate these integrals and will write their generalizations for an arbitrary semi-simple Lie algebra. I will also explain the meaning of asymptotic exactness for Itzykson-Zuber integral and relate it to localization theorems of supergeometry.*Itzykson-Zuber Integral.*

**14/10/2014**Oleg Zaboronski.**Itzykson-Zuber Integral II.****21/10/2014****.**. An introduction to the geometry of Hamiltonian group actions on symplectic manifolds with (counter-)examples (circle actions on the torus and sphere).**Geometric background for the Duistermaat-Heckman integration theorem****28/10/2014**. John Rawnsley.**Geometric background for the Duistermaat-Heckman integration theorem-II.****04/11/2014**. John Rawnsley.**Geometric background for the Duistermaat-Heckman integration theorem-III.****11/11/2014**John Rawnsley**.****. Geometric background for the Duistermaat-Heckman integration theorem-IV.****18/11/2014**Neil O'Connell.**.****Geometric RSK, the Toda lattice and stochastic Backlund transformations -I.****25/11/2014**Neil O'Connell**.****. Geometric RSK, the Toda lattice and stochastic Backlund transformations -II.****02/12/2014**Neil O'Connell.**.****Geometric RSK, the Toda lattice and stochastic Backlund transformations****-III.**

**Term 2**

**13/01/2015.**Mikhail Poplavskyi**.**Toeplitz and Fredholm determinants arise in different areas, and their asymptotic behaviour play crucial role in asymptotic analysis of many integrable systems. We will discuss Szego-Kac-Achiezer type formulas which give an explicit answer in the case of Toeplitz determinants and Fredholm determinants of integral operators with convolution kernels. Also we present some modern results on Fredholm determinants of integral operators with so-called integrable kernels. The talk is based on papers of Szego, Kac, Achiezer, Hirschman, Its, Deift, Krasovsky, Widom, Basor, Tracy. Bibliography*Asymptotic formulae of Szego-Kac-Achiezer type.*

**20/01/2015**. Mikhail Poplavskyi.**Asymptotic formulae of Szego-Kac-Achiezer type-II.****27/01/2015.**Mikhail Poplavskyi*.***Asymptotic formulae of Szego-Kac-Achiezer type-III.****03/02/2015**Mikhail Poplavskyi.**.****Asymptotic formulae of Szego-Kac-Achiezer type-IV.****10/02/2015.**Mikhail Poplavskyi*.***Asymptotic formulae of Szego-Kac-Achiezer type-V.****17/02/2015.**Yuchen Pei.*Quantum Lévy stochastic area and the Catalan numbers.***24/02/2015.**Yuchen Pei.**Quantum Lévy stochastic area and the Catalan numbers-II.****03/03/2015.**Yuchen Pei.**Quantum Lévy stochastic area and the Catalan numbers-III.****10/03/2015.**Tatyana Shcherbina. (*Chebyshev Laboratory, Department of Mathematics and Mechanics of St. Petersburg State University.*). In this talk we will discuss an application of the supersymmetric method (SUSY) to the analysis of the bulk local regime for some types of random band matrices (RBM). SUSY approach is widely used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is difficult and so far for RBM it has been performed only for the density of states. Here we present some rigorous SUSY results about the universality of the local regime of the second correlation function of characteristic polynomials of RBM, as well as about the full second correlation function of the Gaussian block RBM.**Universality of the local regime for some types of random band matrices**

## Term 3

**21/04/2015**No seminar - the first week of term**28/04/2015**Gaetan Borot. (*Max Planck Institute for Mathematics, Bonn.*)Schwinger-Dyson equations form a set of exact constraints on observables in the statistical-mechanics of N particles. For random matrix models with eigenvalues playing the role of the particle's position, they can be used to derive asymptotic expansions of the partition function and of the observables themselves when N >> 1. I will describe the principles of this approach. It is based on a combination of large deviation techniques with the analysis of Schwinger-Dyson equation. It has the advantage to be of purely probabilistic nature, in particular it does not rely on integrability of the model. This is based on joint works with Alice Guionnet and Karol Kozlowski.*Asymptotic expansion in 1d Coulomb gases.*ABSTRACT.**05/05/2015**Nick Simm. (*Warwick*)The characteristic polynomial is a fundamental object in random matrix theory. I will survey a powerful heuristic method to obtain statistics of the extreme values taken by this polynomial, when the matrix size goes to infinity. The results share commonalities with a broad class of logarithmically correlated fields (of which the characteristic polynomial belongs), including branching Brownian motion and the 2D Gaussian Free Field. I will first describe previous results obtained in*Random matrices: Extreme value statistics of the characteristic polynomial*. ASTRACT.

the case of random unitary matrices (the so-called CUE) before describing my recent work (joint with Yan Fyodorov) on the same question for random Hermitian matrices (the GUE) and the new challenges this brings.**12/05/2015**(AT 13:00 - please note the special time!) Nick Simm. (*Warwick)*.**Random matrices: Connections to log-correlated Gaussian fields****ABSTRACT**. I will describe the large-N limit of characteristic polynomials of N x N random matrices from the GUE. In order to get a well-defined limit, an appropriate regularization must be introduced. I will discuss two such regularizations: the first gives the limit as a Gaussian "random distribution", while the second leads to a well-defined stochastic

process related to fractional Brownian motion. Both limits are examples of log-correlated Gaussian fields.**19/05/2015**Kirillov - seminar cancelled**26/05/2015**No seminar - CLAY Institute conference on random polymers**02/06/2015**Mario Kieburg (*Bielefeld*).*Local Spectral Statistics of Products of Random Matrices*.**Abstract.**Products of matrices, in particular of random matrices, is one of the oldest problem of random matrix theory and reaches back into the late 60's. In the past few years this topic experienced a revival due to new analytical results for a certain class of random matrices. I will give a survey of these new developments. In particular I will present two theorems about the exact analytical result for a product of finite dimensional random matrices and roughly sketch their proofs. Moreover I will show you what the behaviour of the spectrum is when the number of matrices multiplied goes to infinity or when the matrix dimension reaches infinity. Thereby I concentrate on the local spectral statistics which is on the scale of the local mean level spacing.**09/06/2015***No seminar***16/06/2015**Chin Lun (*Warwick*).. I will talk about a multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren which can be considered as a hierarchy of partition functions for the continuum directed random polymer. I will discuss the motivation for such an extension and its properties such as continuity, strict positivity and Markov property.*A multi-layer extension of the stochastic heat equation*. Abstract**23/06/2015**Chin Lun (*Warwick*).**A multi-layer extension of the stochastic heat equation-II.**