# 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

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.