# Mathematical Physics and Probability Reading Seminar 2016-17

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

## Term 1

04.10.2016. No seminar

11.10.2016 Luis Carlos Garcia del Molino (University Paris Diderot)

Title: Spectral properties of real non-symmetric random matrices

Abstract: Real non-symmetric random matrices have the surprising property that an unbounded number of eigenvalues accumulate on the real axis. Understanding the respective distributions of real and non-real eigenvalues of the spectrum is challenging and very important for applications in different fields.

In this talk I will first introduce the log-gas technique for the eigenvalues of random matrices and later the specific case of the Real Ginibre ensemble. Using this log-gas I will show that the distribution of the number $k$ of real eigenvalues is asymptotically Gaussian with universal scaling $O(\sqrt{n})$ for their mean and variance. Moreover, I will show that the empirical spectral distribution of real eigenvalues undergoes a transition between bimodal for $k = o (\sqrt{n})$ to unimodal distributions for $k = O (\sqrt{n})$, with a uniform distribution at the transition. Finally I will provide numerical evidence that this results hold for a wide range of random matrices with independent entries beyond the universality class of the circular law.

18.10.2016 Oleg Zaboronski (Warwick)

Title: An introduction to the six vertex model

Abstract: Following Franchini, we will define the six vertex model, introduce the transfer matrix for the model subject to the so-called 'ice condition' and derive the corresponding Yang Baxter equation.

25.10.2016 Oleg Zaboronski (Warwick)

Title: An introduction to the six vertex model-II

Abstract: Following Franchini, we will study the R-matrix for the six vertex model.

01.11.2016 Oleg Zaboronski (Warwick)

Title: An introduction to the six vertex model-III

Abstract: Following Baxter, we will derive Bethe's equations for the eigenvalues of the transfer matrix.

08.11.2016 Oleg Zaboronski (Warwick)

Title: An introduction to the six vertex model-IV

Abstract: In the last talk of the series, we will construct Baxter's Q-operators

15.11.2016 Will Fitzgerald (Warwick)

Title: The Stochastic Six Vertex Model and its connection to interacting particle systems

Abstract: We discuss an equivalent formulation of the stochastic six vertex model as an interacting particle system. Special cases of this include ASEP and TASEP. This talk is based on a paper by Borodin, Corwin and Gorin.

22.11.2016 Roger Tribe (Warwick)

Title: Stochastic 6 vertex model: Borodin, Corwin, Gorin.

Abstract: Part II. The particle system Will introduced has a transition matrix given by the transfer operator for the 6 vertex model. Knowledge of its eigenvalues and eigenvectors has then to be massaged into useful formulae for expectations of observables.

29.11.2016 No seminar

06.12.2016 Roger Tribe (Warwick)

Title: Spectral theory for Particle systems that are solvable by coordinate Bethe ansatz.

Abstract: This is the title of a 2015 paper by Borodin, Borwin, Petrov and Sasamoto
which I find slightly easier to read than the later borodin papers on rational
symmetric functions, and lays out the fourier type identities in a sensible way.
But I won't get far - so only friendly experts are welcome. No probability involved.

References for Term 1

1. Borodin, Petrov. Higher spin six vertex model and symmetric rational functions, arXiv:1601.05770

3. Borodin. On a family of symmetric rational functions, arXiv:1410.0976

4. Franchini. Notes on Bethe Ansatz techniques, full text

5. Jimbo. Introduction to the Yang-Baxter equation, full text

6. Borodin, Corwin and Gorin, Stochastic six-vertex model, arXiv:1407.6729

## Term 2

10.01.2017 No seminar

17.01.2017 Bruce Westbury (Warwick)

Title: The algebraic Bethe ansatz

(solving the six vertex model using the quantum inverse scattering method)

Abstract: In previous talks we have seen that the coordinate Bethe ansatz
solves the Heisenberg spin chain model and the six vertex model
(with periodic boundary conditions). This is based on Bethe's 1932 paper
on the Heisenberg spin chain model. This involves deriving the Bethe
equations and constructing an eigenvector (with eigenvalue) for each
solution.

This is an elementary approach (in the sense that there are very few
prerequisites). However it is difficult to motivate. In his 1982 book
Baxter improved the method by using the Yang-Baxter equation. In this
talk I will present the algebraic Bethe ansatz which is based on the
quantum inverse scattering method. This approach is more sophisticated
(using abstract algebra) and takes the Yang-Baxter equation as the
starting point. The original motivation was to develop a quantum version
of the inverse scattering method. It led directly to the introduction of
the quantum analogue of the coordinate ring of SL(2) and inspired
Drinfel'd's seminal work on quantum groups.

The main reference is [2], KBI, and I have the library copy
on my desk.

24.01.2017 Bruce Westbury (Warwick)

Title: The algebraic Bethe ansatz-II

31.01.2017 Yacine Barhoumi-Andreani (Warwick)

Title: REPRODUCING KERNEL HILBERT SPACES OF SYMMETRIC
FUNCTIONS

Abstract: Building on the previous exposition by R. Tribe, we make more explicit the
formalism at the heart of the Plancherel theory in the case of classical Hilbert spaces of
symmetric functions. This formalism uses the notion of Reproducing Kernel Hilbert Space,
a Hilbert space of functions whose evaluations at a point are continuous for the underlying
topology.
We will remind the main notions and treat the case of the space of square integrable
functions on the unitary group U(N) (endowed with its Haar measure) that are invariant
with respect to conjugacy. The orthogonal functions of this space are the Schur functions and its
reproducing kernel is given by the celebrated Cauchy product (See e. g. Macdonald's book). If

time permits, we will treat the case of the Hall-Littlewood polynomials.

Title: Associative algebras and one-dimensional stochastic processes

Abstract: Some one-dimensional quantum spin chains written in a proper basis
describe stochastic processes. One can use associative algebras and write
the Hamiltonians in terms of the generators of the algebras using special
quotients and representations. Some associative algebras can be Baxterized
which makes the system integrable. The associative algebras depend in
general of one parameter. If this parameter is real, it can be related to
the asymetry of the rates of local processes. If the parameter is on the
unit circle and fixed to a value which makes the associative algebra
a semigroup, one obtains nonlocal stochastic processes with fascinating
properties, We show applications of the Hecke, Brauer and
Birman-Murakami-Wenzel algebras. We stress the symmetries observed for
various quotients of the algebras.

14.02.2017 Yacine Barhoumi-Andreani (Warwick)

Title: REPRODUCING KERNEL HILBERT SPACES OF SYMMETRIC
FUNCTIONS-II

Title: Marcenko-Pastur Law for Kendall's Tau

Abstract: We prove that the empirical eigenvalue distribution of Kendall's Rank correlation matrix converges to the Marcenko-Pastur law, under the assumption that the observations are i.i.d random vectors $X_1, \ldots, X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure.

28.02.2017 NO SEMINAR

07.03.2017 Theodoros Assiotis (Warwick)

Title: Stochastic dynamics for the Hua-Pickrell measures

Abstract: I will describe how to construct a Markov process leaving the Hua-Pickrell measures invariant.

14.03.2017 Theodoros Assiotis (Warwick)

Title: Stochastic dynamics for the Hua-Pickrell measures-II

References for Term 2

How the algebraic Bethe ansatz works for integrable models.
Symétries quantiques (Les Houches, 1995),
149--219, North-Holland, Amsterdam, 1998.

2. Korepin, V. E. ; Bogoliubov, N. M. ; Izergin, A. G.
Quantum inverse scattering method and correlation functions.
Cambridge Monographs on Mathematical Physics. Cambridge University
Press, Cambridge, 1993. {\rm xx}+555~pp. ISBN: 0-521-37320-4;
0-521-58646-1

3. Negro, Stefano .
Integrable structures in quantum field theory.
J. Phys. A 49 (2016), no.~32, 323006, 56~pp.

4. Sklyanin, E. K.
Quantum inverse scattering method. Selected topics.
Quantum group and quantum integrable systems,
63--97, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ,
1992.

## Term 3

02.05.2017 Nick Simm (Warwick)

Title: From random matrix theory to exponentials of Gaussian free fields

Abstract: I will discuss how to make sense of the characteristic polynomial of a random matrix in the limit of infinitely large matrices. This problem turns out to be related to exponentiating the Gaussian free field. I will discuss the work of Webb '14 who established this connection in the L^{2}-phase. If time permits (or in a subsequent lecture) I will discuss recent work with G. Lambert and D. Ostrovsky '16 where we extend the convergence to the whole sub-critical phase by an appropriate regularization of the characteristic polynomial.

09.05.2017 Will FitzGerald (Warwick)

Title: Coalescing and annihilating Brownian motions

Abstract:This talk will introduce coalescing and annihilating Brownian motions and how they can be studied by using dualities to show that they are Pfaffian point processes. This gives a relationship between annihilating Brownian motions and real eigenvalues of the real Ginibre ensemble. It can also be used as a starting point to derive asymptotic properties of these systems such as probabilities of large gaps.

16.05.2017 Nick Simm (Warwick)

Title: From random matrix theory to exponentials of Gaussian free fields-II

23.05.2017 Nikos Zygouras (Warwick)

Title: Understanding determinantal structures around KPZ and its fixed point

Please notice the earlier start time: 12:30-14:00

30.05.2017 Nikos Zygouras (Warwick)

Title: Understanding determinantal structures around KPZ and its fixed point-II

Please notice the earlier start time: 12:30-14:00

06.06.2017 Bruce Westbury (Warwick)

Title: The box ball system.

Abstract:

The box ball system is a combinatorial integrable system. It can be

derived from the Heisenberg spin $1/2$ XXZ spin chain model by taking

$q = 0$. It can also be understood as an ultra-discrete version of the

KdV (Kortweg - de Vries) equation. The discrete KdV equations are
integrable difference equations which are a discrete time version of
the KdV equation. The ultra-discrete equations are an integrable system
in which time and space are discrete and which are a discrete space
version of the discrete KdV equations.

The box ball system has a sequence of commuting operators. These
operators have a simple and elementary definition. There is a
combinatorial correspondence between the states and new objects,
known as rigged partitions. This correspondence gives an induced
action of the commuting operators on rigged partitions. These actions
are linear. This should be understood as explicitly simultaneously
diagonalising the commuting operators.