Monday 11th to Friday 15th September 2023

Department of Mathematics, Pevensey II building, room 5C10

Participants:

• Márton Balázs (Bristol)
• Jacob Butt (Sussex)
• Sunil Chhita (Durham)
• Antoine Dahlqvist (Sussex)
• Elnur Emrah (Bristol)
• Farshid Farshidfar (Sussex)
• Nicos Georgiou (Sussex)
• Jessica Jay (Bristol)
• Janosch Ortmann (Québec)
• Mustazee Rahman (Durham)
• Dominik Schmid (Bonn)
• Nicholas Simm (Sussex)

Funding:

The workshop is supported by the Heilbronn Institute for Mathematical Research.

Programme

 

Time	Monday	Tuesday	Wednesday	Thursday	Friday
10-11am	Nicholas Simm	Elnur Emrah	Sunil Chhita	Jessica Jay	Marton Balazs
11-11.30am	Coffee	Coffee	Coffee	Coffee	Coffee
11.30-12pm	Nicholas Simm – Open problem session	Elnur Emrah– Open problem session	Sunil Chhita– Open problem session	Jessica Jay– Open problem session	Marton Balazs– Open problem session
12-2pm	Lunch	Lunch	Lunch	Lunch	Lunch
2-3pm	Nicos Georgiou	Mustazee Rahman	Jonosch Ortmann	Dominick Schmid
3-3.30pm	Coffee	Coffee	Coffee	Coffee
3.30-4pm	Nicos Georgiou– Open problem session	Mustazee Rahman– Open problem session	Jonosch Ortmann– Open problem session	Dominick Schmid– Open problem session

 

Abstracts

Fluctuations in inhomogeneous last passage percolation

Monday

Nicholas Simm (University of Sussex)

Title: Correlations of characteristic polynomials and last passage percolation

Abstract: Models of last passage percolation have various intriguing connections to random matrix theory. I will review some known examples of this, in particular a formula by Baik and Rains that relates the distribution of the last passage time to correlations of characteristic polynomials. Then I will discuss recent work on the evaluation of such correlations in a broader context of non-Hermitian matrices. The method is based on the use of symmetric function theory, especially Schur functions. This is joint work with my PhD student A. Serebryakov.

Nicos Georgiou (University of Sussex)

Title: Hydrodynamic limit for a TASEP with space-time discontinuous jump rates

Abstract: The totally asymmetric simple exclusion process is a conservative particle system that has been studies though various mathematical lenses. Results for this particle system include hydrodynamic limits, invariant distributions, fluctuations and large deviations.

It has connections to the celebrated KPZ class via a coupling with the corner growth model and last passage percolation; it is considered one of the exactly solvable models of the KPZ class.
In this talk we will discuss a (non-exactly solvable) generalisation of TASEP in which the rates that govern the particle jumps depend on the location of the particle and the time that we are observing the process. The rates come from a background function that can be discontinuous in space and time.
We will discuss the hydrodynamic limit of this version of TASEP (for particle current and density), which will be the solution to certain discontinuous PDEs. This is joint work with Enrico Scalas and Jacob Butt.

Tuesday

Elnur Emrah (University of Bristol)

Title: Right-tail moderate deviations in exponential last-passage percolation

Abstract: We present a soft probabilistic technique for deriving sharp moderate deviation bounds for the right (upper) tail of directed last-passage percolation with exponential weights. This technique utilizes no integrable structure beyond the product-form invariant measures, and can be implemented for various other models within the Kardar- Parisi-Zhang universality class. We will briefly touch on these extensions and some future research directions as well. Based on partly ongoing joint works with C. Janjigian, T. Seppa ̈la ̈inen and Y. Xie.

Mustazee Rahman (Durham University)

Title: Some formulas for inhomogeneous last passage percolation

Abstract: I will describe the exponential last passage percolation model with inhomogeneous weights. The weight at lattice site (i,j) has rate a_i + b_j, where the a_i’s and b_j’s are parameters for inhomogeneity. I will derive formulas for the last passage probabilities of this model and describe their origin through the RSK algorithm. I will then discuss potential applications of these formulas. This is a joint work with Kurt Johansson.

Wednesday

Sunil Chhita (Durham University)

Title: GOE Fluctuations for the maximum of the top path in ASMs

Abstract: The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling and will discuss many open problems related to this model. This talk is based on joint work with Arvind Ayyer and Kurt Johansson.

Janosch Ortmann (UQAM)

Title: The coupling method for central moment bounds in exponential last-passage percolation

Abstract: The probabilistic coupling approach to study stochastic planar models originated in the works of  Cator and Groeneboom on Hammersley’s process and the Poisson last-passage percolation (LPP). It has since been developed to treat many aspects of the KPZ universality in various models of directed LPP, directed polymers and interacting particles.

Based on joint work with Elnur Emrah and Nicos Georgiou, I will discuss how the coupling framework can be used to derive optimal-order central moment bounds in exponential LPP with and without boundary.

Thursday

Jessica Jay (University of Bristol)

Title: Blocking Measures for Interacting Particle Systems and Combinatorial Identities 

Abstract: Recent research has found that studying reversible stationary measures, known as blocking measures, for exclusion type interacting particle systems can lead to probabilistic proofs and interpretations of both well-known and new identities of combinatorial significance. 

In 2018, Balázs and Bowen gave an elegant probabilistic proof of the Jacobi Triple Identity; a fundamental sum-product identity with interpretations throughout Mathematics and Physics. The proof follows by considering the equivalence between ASEP and the Asymmetric Zero-Range Process.  In 2022, Balázs, Fretwell and Jay generalised this framework by considering a parameterized family of 0-1-2 particle systems under their blocking measures and found an equivalent family of kinetically constrained systems. By doing this they give probabilistic proofs to new three variable combinatorial identities. 

More recently, this year, Adams, Balázs and Jay gave probabilistic proofs and interpretations to combinatorial identities due to Euler, Gauss, and Heine, by considering very natural distributional questions for ASEP. 

In this talk, we will review these probabilistic proofs and the connection between particle systems and combinatorial structures.

Dominik Schmid (University of Bonn)

Title: Approximating the stationary distribution of the ASEP with open boundaries

Abstract: In this talk, we study the stationary distribution of asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment, and provide conditions on the length and size of the subinterval such that this projection is close in total variation distance to a product measure. In particular, we discuss the special case of the TASEP with open boundaries, where the approximation of the stationary distribution can be reformulated as a question about coalescence of geodesics in inhomogeneous last passage percolation. 

This talk is based on joint work with Evita Nestoridi. 

Friday

Marton Balazs (University of Bristol)

Title: Road layout in the KPZ class

(growing out of a project that started with Riddhipratim Basu, Atal Bhargava, Sudeshna Bhattacharjee, Karambir Das, Sanchari Goswami, David Harper, Sunil Kumar, Aquib Molla)

Abstract: In this talk I'll be after a model for road layouts. Imagine a Poisson process on the plane for start points of cars. Each car picks an independent random direction and goes straight that way for some distance. I'll start with showing that the origin (my house, that is) will see a lot of car traffic in an arbitrary small distance.

Which is not what we find in the real world out there, why? The answer is of course coalescence of paths in a random environment provided by hills and other geographic and societal obstacles. Which points towards first passage percolation (FPP), expected to be a member of the KPZ universality class. Due to lack of results for FPP, instead we build our model in exponential last passage percolation (LPP), known to be in the KPZ class. It's quite clear what the model should be, I'll define it anyway and discuss some results about road geometries in this LPP model. I'll also mention our current struggle trying to match our theorems with actual traffic data.

Travel

The University of Sussex is located in Falmer which is a short bus or train ride away from Brighton, an hour from London and 30 minutes away from Gatwick international airport. (directions to the University).

Further local information is available at Visit Brighton (webpage image supplied by VisitBrighton)

Campus map