# Mathematical Physics and Probability Reading seminar

The Mathematical Physics and Probability seminar will be running during Winter 2021 on Tuesdays at 4:30pm via zoom (link to be provided via email).

**Term 3.**

**Schedule**

**27/4 Evgeni Dimitrov** (Columbia)

**Title:** Towards universality for Gibbsian line ensembles

**Abstract:** Gibbsian line ensembles are natural objects that arise in statistical mechanics models of random tilings, directed polymers, random plane partitions and avoiding random walks. In this talk I will discuss a general framework for establishing universal KPZ scaling limits for sequences of Gibbsian line ensembles. This framework is still being developed and I will explain some of the recent progress that has been made towards carrying it out for two integrable models of random Hall-Littlewood plane partitions and log-gamma polymers.

**4//5 Mustazee Rahman **(Durham)

**Title:** A random growth model and its time evolution

**Abstract:** Planar random growth models are irreversible statistical mechanical systems with a notion of time evolution. It is of some interest to understand this evolution by studying joint distribution of points along the time-like direction. One of these models, polynuclear growth or last passage percolation, has allowed exact statistical calculations due to a close relation to determinantal processes. I will discuss recent works with Kurt Johansson where we calculate its time-like distribution using ideas surrounding determinantal processes. One can take a scaling limit of this distribution, which is then expected to be universal for the time distribution of many random growth models.

**11/5 Dan Bump** (Stanford)

**Title:** Colored Lattice Models and Whittaker Functions

**Abstract: **Two seemingly different constructions, one from number theory and the other from mathematical physics, are Whittaker functions of p-adic groups, and solvable lattice models.Whittaker functions are special functions on a group such as $GL(n,F)$, where $F$ is a $p$-adic field, or a metaplectic cover of such a group. They play an important role in the theory of automorphic forms.

Solvable lattice models on the other hand are statistical mechanical systems such as the "six-vertex model" proposed by Linus Pauling in 1935 as a model for ice. Thermodynamic properties of the model are controlled by the "partition function", a global sum over states. "Solvability" is a property of particular models that may be studied by algebraic methods based on the Yang-Baxter equation.

We will show how Iwahori Whittaker functions on metaplectic covers of $GL(n,F)$ may be represented as partition functions of solvable lattice models, and how the Yang-Baxter equation leads to the identical Demazure-Whittaker operators that also come from the representation theory of p-adic groups. This is joint work with Brubaker, Buciumas and Gustafsson. See arXiv:1906.04140 and arXiv:2012.15778.

**18/5 Andrew Kels** (SISSA)

**Title: **The star-triangle relation, integrable models, and hypergeometric integrals.**
Abstract: **The star-triangle relation is a special form of the Yang-Baxter equation for two-dimensional Ising-type lattice models of statistical mechanics. This relation implies that the transfer matrices of a lattice model commute, which in principle allows to find an exact solution of the model, that is, to solve for the partition function in the thermodynamic limit. Recently there has also been discovered interesting connections between the star-triangle relation, hypergeometric integrals, and discrete integrable soliton equations. Through this connection most of the important integrable lattice models of statistical mechanics, as well as the discrete soliton equations in the Adler, Bobenko, and Suris classification, can be obtained through limits of certain solutions of the star-triangle relations that are related to elliptic hypergeometric functions. This talk is aimed to be an introduction to the above results regarding the star-triangle relations.

**25/5 Giuseppe Cannizzaro**

**Title: **From the KPZ equation to the KPZ Fixed point: an overview of Quastel-Sarkar's work

**1/6 Simon Gabriel**

**Title:** Comparing transition probabilities of two Markov processes

**8/6 Sigurd Assing**

**Title: **On the strong sector condition and applications to interacting particle systems I

**15/6 Sigurd Assing**

**Title: **On the strong sector condition and applications to interacting particle systems I

**22/6 Anna Puskas **(Queensland)

**Title: ** Demazure-Lusztig operators and Metaplectic Whittaker functions**
Abstract: **The study of objects from number theory such as metaplectic Whittaker functions has led to surprising applications of combinatorial representation theory. Classical Whittaker functions can be expressed in terms of symmetric polynomials, such as Schur polynomials via the Casselmann-Shalika formula. Tokuyama's theorem is an identity that links Schur polynomials to highest-weight crystals, a symmetric structure that has interesting combinatorial parameterisations.

In this talk, we will review the significance of metaplectic Whittaker functions, and discuss them in the context of representations of the Iwahori-Hecke algebra. We will examine combinatorial constructions for these objects inspired by Tokuyama's theorem. This theory carries over to the infinite-dimensional setting, and connects with work on double affine Hecke algebras, where several intriguing open questions remain

**.**

**29/6 Tommaso Rosati **(Imperial) (**Special time: 11:30am**)

**Title:** Convergence to the KPZ fixed point with finite energy initial datum

**Term 2. **

In this term we will be looking at important, recent developments on multi-correlation convergences of KPZ models. Some relevant references are (the list is not exhaustive): Heat and Landscape, Directed Landscape, KPZ fixed point I, KPZ fixed point II, Stochastic 6-vertex.

**Schedule**

**19/01 Duncan Dauvergne **(Princeton), **SPECIAL TIME 4pm**

**Title:** The Airy sheet and the directed landscape

**Abstract:** The directed landscape is a random `directed metric' that arises as the full scaling limit of last passage percolation, recently constructed by myself, Janosch Ortmann and Balint Virag. In this talk I will try to explain the key new ideas that underlie this construction. The main obstacle is constructing the Airy sheet, a two-parameter scaling limit of last passage percolation, when both the start and end point are allowed to vary spatially. I will describe how the Airy sheet is built from asymptotic last passage values along parabolas in the Airy line ensemble via an isometric property of the RSK bijection.

**26/01, 11am Nikos Zygouras**

**Title:** "RSK aspects of the directed landscape"

**Abstract: **Following Dauvergne's talk last week, I will present a key property of the Robinson-Schensted-Knuth (RSK) correspondence, which relates last passage percolation to a last passage percolation on the Airy line ensemble.

**2/2, 4:30pm, Axel Saenz**

**Title:**Determinantal transition kernels for some interacting particles on the line.

**Abstract:**Based on the 2008 paper by Dieker and Warren with the same name, we couple a pair of Markov processes via the RSK correspondence so that one process has a Karlin-McGregor type kernel and compute the kernel of the second process by an intertwining relation for both kernels.

**9/2, 4:30pm, Sourav Sarkar (Toronto)**

**Title:**Convergence of exclusion processes and the KPZ equation to the KPZ fixed point.

**Abstract:**We will describe a method of comparison with TASEP which proves that both the

present time a class of random initial data, dense in continuous functions. We will give a little background, but the talk will mostly be about the proof. Joint work with Jeremy Quastel.

**16/2, 4:30pm, Ofer Busani (Bristol)**

**Title:**Convergence to the Airy sheet via the Baik-Ben Arous-Peche distribution

**Abstract:**We will go over the main results of the recent paper by Virag - https://arxiv.org/abs/2008.07241, and discuss the main ideas behind some of the proofs in the paper.

**23/2, 4:30pm, Theo Assiotis (Edinburgh)**

**Title:**The heat and the landscape continued

**Abstract:**I will continue from Ofer's talk, after recalling a few things, the discussion of Balint Virag's paper https://arxiv.org/abs/2008.07241.

**2/3, 4:30pm, Axel Saenz**

**Title:**Last passage as a linear combination of edge processes

**Abstract:**We present Theorem 5 of Virag's "The heat and the landscape" based on a previous results of Dieker and Warren (2008).

**9/3, 4:30 pm Jhih-Huang Li**

**Title**: Bounds on the top curve of non-intersecting Brownian motions

**Term 1**

The topic in term 1 will be on Hecke Algebras and applications to integrable probability.

**20/10 Oleg Zaboronski**

**Title:** Reaction-diffusion, exclusion models and Hecke algebras - notes

**Abstract:**Let us consider a class of interacting particle systems on Z, which have a generator where are linear operator, which only act on sites . As it turns out, one can construct a full set of duality functions for such a model, provided g_i's generate Hecke algebra. The models singled out by this condition include ASEP, symmetric annihilating coalescing random walks, branching-coalescing random walks, annihilating random walks with immigration. Will the knowledge of representation theory prove useful in studying these models?

**27/10 Oleg Zaboronski**

**Title:** Reaction-diffusion, exclusion models and Hecke algebras: Part II.

**3/11 Nikos Zygouras**

**Title:** On symmetries of coloured models and Hecke algebras

**Abstract: **I will expose parts of a paper by Pavel Galashin arxiv.2003.06330

**10/11 Nikos Zygouras**

**Title:** On symmetries of coloured models and Hecke algebras II

**Abstract: **I will continue on the exposition on coloured models and Hecke algebras parts following the paper by Pavel Galashin.

**17/11 Axel Saenz-Rodriquez **

**Title:**Coxeter Group Actions on Interacting Particle Systems, I - notes

**Abstract:**We present a color-position symmetry result, first established by Amir-Angel-Valko '11, for the colored asymmetric simple exclusion process (ASEP). The result may be obtained by identifying the configuration of the colored ASEP with elements of the Coexeter group (i.e. the symmetric group on letters). We explain how this approach allows us to extend the result to a general setting with open boundaries and multiple particles per site. The talk is based on the recent paper "Coxeter Group Actions on Interacting Particle Systems" by Jeffrey Kuan.

**23/11 Axel Saenz-Rodriquez **

**Title:**Coxeter Group Actions on Interacting Particle Systems, II, slides

**Abstract:**We present a color-position symmetry result, first established by Amir-Angel-Valko '11, for the colored asymmetric simple exclusion process (ASEP). The result may be obtained by identifying the configuration of the colored ASEP with elements of the Coexeter group (i.e. the symmetric group on letters). We explain how this approach allows us to extend the result to a general setting with open boundaries and multiple particles per site. The talk is based on the recent paper "Coxeter Group Actions on Interacting Particle Systems" by Jeffrey Kuan

**1/12 Jhih-Huang Li**

**Title:**Duality functions for multi-species ASEP via non-symmetric Macdonald polynomials, slides

**Abstract**The talk is based on a paper of Chen, de Gier and Wheeler (arXiv1709.06227). In the first part of the talk, we introduce the notion of duality function and its matrix formalism. Then we present a general method to construct such functions in multi-species asymmetric exclusion processes (mASEP). We will need to make a detour to non-symmetric Macdonald polynomials and explore its properties (singularities and coefficients) which will be the key ingredient in this construction. However, this does not provide us with explicit formulas for duality functions in general, but in some special cases, with the help of the matrix-product Ansatz studied earlier in Cantini, de Gier and Wheeler (arXiv1505.00287), one is able to recover previously-known duality functions for the single-species ASEP.

**8/12 Jhih-Huang Li**

**Title:**Duality functions for multi-species ASEP via non-symmetric Macdonald polynomials, II, notes

**Abstract**The talk is based on a paper of Chen, de Gier and Wheeler (arXiv1709.06227). In the first part of the talk, we introduce the notion of duality function and its matrix formalism. Then we present a general method to construct such functions in multi-species asymmetric exclusion processes (mASEP). We will need to make a detour to non-symmetric Macdonald polynomials and explore its properties (singularities and coefficients) which will be the key ingredient in this construction. However, this does not provide us with explicit formulas for duality functions in general, but in some special cases, with the help of the matrix-product Ansatz studied earlier in Cantini, de Gier and Wheeler (arXiv1505.00287), one is able to recover previously-known duality functions for the single-species ASEP.

**15/12 Bruce Westbury**

**Title:**The Yang-Baxter equation and Richardson varieties

**Abstract:**This is a sequel to Nikos' talk on the stochastic coloured vertex model.

###### References

Below is some preliminary reference list. This list will be updated and refined based on the material and topic that we will decide to focus on.