# 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 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
KPZ equation and finite range exclusion models converge to the KPZ fixed point under the 1:2:3 scaling.
For the KPZ equation and the nearest neighbour exclusion (ASEP), the initial data is allowed to be a continuous
function plus a finite number of narrow wedges, but for non-nearest neighbour exclusions, one needs at 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 $L=\sum_i g_i,$ where $g_i$are linear operator, which only act on sites $(i,i+1)$. 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 $A_{N-1}$ (i.e. the symmetric group on $N$ 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 $A_{N-1}$ (i.e. the symmetric group on $N$ 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.
I will start by reviewing theYang-Baxter basis of the Hecke algebra. This model
is studied in arXiv:2003.06330 by Galashin. The main result of this paper is the
flip theorem. The paper ends with an interpretation of this theorem
in terms of the geometry of flag manifolds. I aim to give an overview of the background
to this. However I will take this as a working seminar and not assume any prior
knowledge.
###### 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.

A.P. Isaev, O.V. Ogievetsky

Chen-de Gier-Wheeler

Borodin-Wheeler

P. Galashin