# Mathematical Physics and Probability Reading seminar 2015-16

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

## Term 1

### 13. 10.15. Yacine Barhoumi (Warwick).

TITLE: SOME APPLICATIONS OF THE BETHE ANSATZ.

ABSTRACT: The Bethe ansatz is a method of computation that allows to derive closed-form expressions for the eigenvalues and eigenvectors of particular quantum many-body interaction systems whose hamiltonians share certain integrability properties. Recently, this method of resolution was applied to probabilistic models such as ASEP to derive determinantal formulas for probabilities.

The goal of this talk is to give a historical introduction to the Bethe ansatz. We will thus present the original form used by Bethe, nowadays known under the denomination of “coordinate Bethe ansatz”. We will then illustrate the relevance of the method to the case of the ASEP.

### 20.10.15. Yacine Barhoumi (Warwick).

TITLE: SOME APPLICATIONS OF THE BETHE ANSATZ-II.

### 27.10.15. Yacine Barhoumi (Warwick).

Title: SOME APPLICATIONS OF THE BETHE ANSATZ-III.

### 03.11.15. Ioanna Nteka (Warwick).

TITLE: The Bethe Ansatz for systems with reflections.
ABSTRACT: I will present the Bethe Ansatz method for systems with reflections. I will show, for example, how one can obtain
solution to the Schrodinger equation on the half-line. I will then present the method for systems associated with
semi-simple groups. The talk is based on the book of M. Gaudin "The Bethe wavefunction".

#### 17.11.15. Bruce Westbury (Warwick).

Title: An introduction to the Yang-Baxter equation.

Abstract: I plan to start by presenting the Yang-Baxter equation. I will then address the two basic questions:

(i) What are the basic examples?

(ii) Why is this equation useful?

The basic example is the single bond transfer matrix in the six vertex model. This equation is useful because it implies that any two row-to-row transfer matrices on a rectangular lattice with periodic boundary conditions commute. This means that there is a basis which simultaneously diagonalises all members of this family. This basis also diagonalises the Hamiltonian of the Heisenberg spin 1/2 XXZ spin chain model and the generator of the ASEP Markov process.

### 24.11.15. Bruce Westbury (Warwick).

Title: An introduction to the Yang-Baxter equation-II.

#### 01.12.15 Mihail Poplavskyi (Warwick).

Title: Stochastic Six Vertex Model and its connections to exclusion processes

Abstract: We discuss Stochastic Six Vertex model as an example of integrable model. We will see how one can obtain some particular exclusion processes as a subcase of the model. The talk is based on a series of papers by A. Borodin, I. Corwin, V. Gorin, L. Petrov, A. Povolotsky.

References

 A. Borodin, I. Corwin, V. Gorin - Stochastic Six-Vertex Model arxiv:1407.67629
 I. Corwin, L. Petrov - Stochastic higher spin vertex model on the line arxiv:1502.07374
 A. Borodin - On a family of symmetric rational functions arxiv:1410.0976

### 08.12.15 Mihail Poplavskyi (Warwick).

Title: Stochastic Six Vertex Model and its connections to exclusion processes -II: higher spin stochastic six vertex model

Abstract: We discuss a generalization of Stochastic Six Vertex Model. We introduce the model where vertices with vertical multilines are allowed and the corresponding weights.
R-matrix and transfer matrix are introduced and the general scheme of solving the model is given. The talk is based on a lecture notes of A.Borodin given at Les Houches.

## Term 2

### 12.01.16 Nicholas Simm (Warwick)

Title: From Bethe Ansatz to determinants for the q-TASEP model

Abstract: We will show how to use Bethe ansatz to obtain determinantal representation for correlation functions in the q-TASEP model. The talk is based on a self-contained presentation given in http://arxiv.org/abs/1207.5035 and http://arxiv.org/abs/1308.3475. The method for finding eigenfunctions given there is similar to the what is used in the analysis of other solvable models including the higher spin vertex model.

### 19.01.16 Ioanna Nteka (Warwick)

Title: Dynamics above a wall

Abstract: I will present a two-dimensional model where each particle evolves as a Brownian Motion with continuous drift that depends on how far the process is from being inside the symplectic Gelfand Tsetlin cone. I will prove that, under appropriate initial condition, each level evolves as an analogue of the Whittaker process with wall.
The talk is based on joint, unpublished work with Dr Warren and Dr Zygouras.

### 26.01.16 Ioanna Nteka (Warwick)

Title: Dynamics above a wall - II

Abstract: I will conjecture a spectral decomposition theorem for the type-B Whittaker functions and recover the entrance law for the Whittaker process with wall. Then I will discuss some main steps for the proof of the theorem.

### 02.02.2016 Yacine Barhoumi (Warwick)

Title: Introduction to the Fock space with probabilistic applications

Abstract: The fermionic Fock space is a natural vector space whose basis is indexed by partitions of an integer, or equivalently Maya diagrams, sequences of 0-1 and normally ordered wedge products. The bosonic Fock space is a vector space of polynomials in infinitely many variables. Each of these spaces were initially considered by particle physicists as a model of a state space where two types of elementary particles interact, the bosons and the fermions.

The mathematical covering of these a priori unrelated spaces uses the language of representation theory. Fermionic Fock spaces are representations of a Clifford algebra whose generators correspond to adding or removing a particle in a given energy state, and bosonic Fock spaces are in the same way representations of a Weyl algebra. These spaces where shown to have a richer structure that we will discuss if time permits : they carry representations of many more algebras such as infinite rank matrix algebras, affine Kac-Moody algebras and quantum groups. Understanding the various relations between the actions of these various algebras on the Fock spaces has proven useful in many applications, and in particular in probability theory with Okounkov’s calculation of the correlation functions of the Schur measure.

The goal of this talk is to define the Fock spaces and their relationship (boson-fermion correspondence) and to show how they can be used to compute quantities of interest in probabilistic particles systems.

## 09.02.2016 Yacine Barhoumi (Warwick)

Title: Introduction to the Fock space with probabilistic applications. Part II.

## 16.02.2016 Yacine Barhoumi (Warwick)

Title: Introduction to the Fock space with probabilistic applications. Part III.

## 23.02.2016 Yacine Barhoumi (Warwick)

Title: Introduction to the Fock space with probabilistic applications. Part IV.

## 01.03.2016 Jan Felipe van Diejen (Universidad de Talca)

Title: Bethe Ansatz for a finite q-boson system with boundary interactions

Abstract:

We construct an orthogonal basis of algebraic Bethe Ansatz eigenfunctions for a finite q-boson system endowed with diagonal open-end boundary interactions.
Via a continuum limit, this allows to verify the orthogonality of the Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove with repulsive Robin boundary conditions.

## 08.03.2016 Mario Kieburg (University of Bielefeld)

Title: Products of Random Matrices: What is new?

Abstract: Products of random matrices were first studied in the late 60's, early 70's to analyze the stability of time evolutions in complex systems. Since then other applications as in transport theory, quantum information, quantum chromodynamics, etc. were found. Especially for the newer applications new challenges arose for this kind of random matrix ensembles. In the past years a new radical progress was achieved in the derivation of the spectral statistics of product matrices at finite matrix dimensions. I will review this development for a certain class of random matrices multiplied from the point of view of harmonic analysis. For this purpose I will recall the class of polynomial ensembles and a distinguished subclass therein. Moreover I will show how the kernels for the spectral statistics of the eigenvalues and of the singular values change when multiplying the corresponding random matrices.

## 15.03.2016 Gernot Akemann (University of Bielefeld)

Title: Products of random matrices: dropping the independence

Abstract: The subject of products of independent random matrices has seen considerable progress recently. This is due to the observation that its complex eigenvalues and its singular values form determinantal point processes that are integrable. In this talk I will drop the independence and present results for the singular values of the product of two linearly coupled random matrices as a first step. The resulting ensemble is no longer polynomial, but can be solved explicitly in terms of an integrable kernel of biorthogonal functions. The process interpolates between that of two independent and a single Gaussian random matrix. For genuine coupling in the local scaling regime at the origin we find back the Meijer G-kernel of Kuijlaars and Zhang which is universal. Sending the coupling to zero at a rate 1/N, where N is the matrix size, we find a one-parameter family of kernels that interpolates between the Bessel- and Meijer G-kernel for two independent matrices. In this limit the two matrices in the product become strongly coupled and almost adjoint to each other. This is joint work with Eugene Strahov.

## 17.05.2016 Yacine Barhoumi (Warwick)

Title: On the order of the maximum of the branching random walk

Abstract: The Branching Random Walk (BRW) is a natural model that describes the
evolution of a population of particles in the presence of a spatial movement. It was
introduced independently in the 30ies by Kolmogorov-Petrovsky-Piskounov and Fisher.
Approximate branching structures recently regained a particular interest in the study
of the maximum of the characteristic polynomial of a random Haar-distributed unitary
matrix and the Riemann Zeta function on the critical line.
This talk aims at explaining the general strategy to obtain the order of the maximum
of the Branching Random Walk.

## 24.05.2016 Yacine Barhoumi (Warwick)

Title: On the order of the maximum of the branching random walk, Part II.

## 31.05.2016 Dirk Erhard (Warwick)

Title: An introduction to the Gaussian multiplicative chaos

Abstract: The gaussian multiplicative chaos is a random measure with density equal to the exponential of a gaussian field with respect to some Radon measure.

It becomes particularly interesting when the covariance function of the gaussian field turns out to exhibit singularities on the diagonal so that the field is not well defined as a random function but is a generalized function/distribution instead.

An example to keep in mind is the two dimensional gaussian free field.

In this case the notion of gaussian multiplicative chaos is not well defined and I will explain in this talk how to still give a meaning to it and I try to explain that this meaning is canonical in a certain sense.

## 07.06.2016 Yacine Barhoumi (Warwick)

Title: Gaussian multiplicative chaos continued
Abstract: We will build on the previous talk by Dirk and give some properties of the
Gaussian Multiplicative Chaos. When the class of measures that is used in the definition
of the GMC has a certain type of regularity, we will see that the associated GMC inherits
from this regularity. This implies the existence of a critical parameter up to which the
GMC is non zero. If time permits, we will give some properties of the critical GMC when
the covariance of the integrated Gaussian field is logarithmic.

## 14.06.2016 Pavlos Tsatsoulis (Warwick)

Title: The Stochastic Quantization Equations on the 2-dimensional torus. Existence of Invariant Measures
Abstract: See abstract.

## 21.06.2016 Mihail Poplavskyi (Warwick)

Title: Applications of orthogonal polynomials for studying random unitary matrices
Abstract: We start with defining a Laurent family of trigonometric polynomials orthogonal with respect to some weight. This family satisfy 5-term recurrent relation with the coefficients called Verblunsky coefficients. We then discuss connections between asymptotic properties of Verblunsky coefficients and some standard problems in RMT.

## 28.06.2016 Mihail Poplavskyi (Warwick)

Title: Applications of orthogonal polynomials for studying random unitary matrices-II
Abstract: We finish with the derivation of 5-term recurrent relations for orthogonal Laurent Polynomials. Some important properties of corresponding CMV matrices will be given. We then show how to use all the above to prove universality at the edge for unitary matrix models and to derive 5 diagonal matrix model for unitary beta ensembles.

## 05.07.2016 Joseph Najnudel (Toulouse)

Title: On the maximum of the characteristic polynomial of the Circular Beta Ensemble.
Abstract: The Circular Beta Ensemble corresponds to a random set of N points on the unit circle whose probability density is proportional to the product of the mutual distances between the points, to the given power beta. If this power is equal to 2, this ensemble can be obtained by taking the eigenvalues of a Haar-distributed random matrix on the unitary group U(N). In a joint work with Chhaibi and Madaule, we show that the logarithm of the maximal modulus, on the unit circle, of the random monic polynomial whose zeros are the points of the Circular Beta Ensemble, is equal to sqrt(2/beta) (log N - (3/4) log log N + X_N), where X_N (N larger than 2) form a tight family of random variables. Our result partially solves a conjecture by Fyodorov, Hiary and Keating. It improves, and generalizes to all values of beta, some recent results by Arguin, Belius, Bourgade, Paquette and Zeitouni, available for beta equal to 2.