# 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 13:00, B3.02**

## Term 1

**2 October.** No seminar

**9 October.** Leonid Petrov (Virginia). *Cauchy identities, Yang-Baxter equation, and their randomization*.

**Abstract.** Cauchy type summation identities for various families of symmetric polynomials (with Schur polynomials as the first example) are crucial in bringing exact solvability to various stochastic particle systems in the Kardar-Parisi-Zhang universality class. First breakthroughs in this direction about 20 years ago employed Robinson-Schensted-Knuth correspondences to study asymptotic fluctuations of longest increasing subsequences and TASEP (totally asymmetric simple exclusion process). Deforming the Schur structure, one can connect Cauchy identities to the Yang-Baxter equation for the six vertex model, and use this to exactly solve more general models such as ASEP. I will discuss how the structure of RSK correspondences should be adapted in connection with these deformations, providing a "bijective" point of view on the Yang-Baxter equation.

**16 October.** Jon Warren (Warwick). *A first look at the Gaussian Free Field.*

**Abstract**. I will try to give some intuition for why this is a fundamental process, starting with a discrete version, emphasizing the Markov property, and concluding with a quick look at an example of an interesting model in which the GFF arises.

**23 October. ** Jon Warren and Sigurd Assing (Warwick). *A first look at the Gaussian Free Field-II.*

**Abstract .** Construction of a Gaussian measure whose covariance kernel is given by the inverse of the Laplacian on a bounded domain with Dirichlet boundary condition a la L. Gross's theory of Abstract Wiener Spaces.

**30 October**. Sigurd Assing (Warwick). *The Gaussian Free Field-III*.

**Abstract**. Construction of a Gaussian measure whose covariance kernel is given by the inverse of the Laplacian on a bounded domain with Dirichlet boundary condition a la L. Gross's theory of Abstract Wiener Spaces.

**6 November**. Oleg Zaboronski (Warwick). *Two-dimensional Green's functions.*

**Abstract**. To prepare for the study of special points of GFF, we review basic properties of 2d Green's functions: conformal invariance

and the asymptotic behaviour near the singularity.

**13 November**. Sigurd Assing (Warwick).* Circle Averages and Markov Property of GFF.*

**Abstract**. I will discuss circle averages but also revisit the Markov property introduced by Jon and try to connect it to a sharp Markov property. The main target is to prepare for the construction of the Liouville measure.

**20 November.** *No seminar*

**27 November.** Oleg Zaboronski (Warwick). *GFF: conformal invariance and the set of thick points.*

**Abstract.** We will go through sections 1.9-1.11 of Nathaniel's notes. The hope is to define circle averages, relate them to Brownian motions

and study (heuristically to start with) the set of $\alpha$-thick points of GFF.

**4 December.** Kurt Johansson (KTH). *Understanding the two-time distribution in local random growth.*

**Abstract:** There has been recent progress on the two-time distribution in

certain local random growth models related to directed last-passage percolation

in two dimensions. In previous work, I have derived a rather complicated formula

for the limiting two-point distribution in the form of an integral of a Fredholm

determinant. In the talk, I will give some background and discuss how you can

get some interesting information out of the formula for the two-time distribution.

## Term 2

**08 January**. No seminar

**15 January**. Roger Tribe (Warwick). * Liouville measure.*

**Abstract**: Following Chapter 2 of Berestycki's notes, I will assign a rigorous meaning to the exponential of the Gaussian free field, an object of fundamental importance for the theory of random surfaces.

**22 January**. Roger Tribe (Warwick). *Liouville measures-II.*

**Abstract:** Following Chapter 2 of Berestycki's notes, I will assign a rigorous meaning to the exponential of the Gaussian free field, an object of fundamental importance for the theory of random surfaces.

**29 January**. Jacek Kiedrowski (Warwick). *Knizhnik-Polyakov-Zamolodchikov formula. *

**Abstract**: Following Chapter 3 of Beresticki's notes, I will introduce the KPZ formula which is closely linked with computing critical exponents of models of statistical physics.

**05 February.** Jacek Kiedrowski (Warwick). *Knizhnik-Polyakov-Zamolodchikov formula-II.*

**Abstract:** Following Chapter 3 of Beresticki's notes, I will recap the KPZ formula and prove the case of expected Minkowski dimension using the multifractal spectrum of Liouville measure.

**12 February.** Will FitzGerald (Warwick). *Random planar maps and Liouville quantum gravity.*

**Abstract:**I will try to give the intuition behind the connections (many of which are conjectural) between random planar maps and Liouville quantum gravity. This is following Chapter 4 of Berestycki's notes.

**19 February.** Will FitzGerald (Warwick). *Scale-Invariant Random Surfaces.*

**Abstract:**I will construct the free boundary Gaussian free field and quantum wedges. This is following Chapter 5 of Berestycki's notes

**26 February**. Will FitzGerald (Warwick). *Scale-Invariant Random Surfaces-II.*

**05 March.** No seminar

**12 March**. Jon Warren (Warwick). *The Gaussian free field and SLE.*

**Abstract.** I will discuss chapter 6 of Berestycki’s notes and the paper by Sheffield, Conformal weldings of random surfaces: SLE and the quantum gravity zipper.

When two (appropriately chosen) quantum surfaces are ( appropriately ) joined together, they result in a new quantum surface containing the random line along which the original surfaces are joined. That random line, amazingly, turns out to be described by an SLE curve.

## Term 3

**07 May**. Roger Tribe (Warwick). *Parisi Soulas dimensional reduction,1982.*

**Abstract. **This work has not been properly appreciated by the probabilists (though David Elworthy ran an M.Sc. module on it at the time). The results can be stated as exact (but unexpected) identities about the (partial) law of certain non-linear elliptic SPDE. A recent pre-print by Albevario, de Vecchi and Gubinelli tries to advertise (and confirm) the results. I will try and give a quick overview of what is in this paper in the hope we might want to look at details.

**21 May**. Will FitzGerald (Warwick). *Directed polymers in the KPZ universality class*

**Abstract.**I will discuss an equality in distribution between the point-to-line partition functions of two directed polymers, with an inverse-Gamma and a Brownian environment. This generalises Dufresne's identity.

**4 June**. Dom Brockington (Warwick).

*Large deviations for sticky Brownian motions*

**Abstract**. I will present some recent results of Barraquand and Rychnovsky on Tracey-Widom fluctuations for a class of stochastic flows of kernels describing systems of Brownian motions with sticky interactions. These results can also be interpreted in terms of the largest particle of the sticky Brownian motions.

**Special seminar: 20 June, 10:00-11:00 at MS.04**. Dom Brockington (Warwick).

*Large deviations for sticky Brownian motions-II*

**Reading List**

N. Berestycki, Introduction to the Gaussian Free Field and Liouville Quantum Gravity

W. Werner, Topics on the two-dimensional Gaussian Free Field

O. Zeitouni,Gaussian Fields

Marek Biskup*, *Extrema of 2D Discrete Gaussian Free Field

Michael Kozdron, Basic theory of univalent functions