Integrable probability mini-workshop

In conjunction with Ivan Corwin's course, there will be a number of talks that week on related topics. Tentative schedule (in progress):

Schedule Speaker Title Links
June 11, 16:00 Duncan Dauvergne The Airy Sheet
June 11, 16:50 Milind Hegde
Jacob Calvert 		The quantitatively Brownian nature of the Airy line ensemble
June 11, 17:30 Erik Bates Endpoints of disjoint geodesics in the directed landscape
June 11, 18:00 Shirshendu Ganguly Geodesic watermelons in last passage percolation
June 12, 16:00 David Croydon Invariant measures for KdV and Toda-type discrete integrable systems slides
June 12, 16:50 Xuan Wu Tightness of the KPZ line ensemble slides
June 12, 17:10 Sourav Sarkar Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition
June 12, 17:30 Lingfu Zhang Empirical distribution along geodesics in exponential last passage percolation slides
June 12, 18:00-18:40 Sylvie Corteel Multitype ASEP and Macdonald polynomials slides

Abstracts

Duncan Dauvergne: The Airy Sheet

The Airy sheet is a random two-parameter function that arises as the scaling limit of last passage percolation when both the start and end point are allowed to vary spatially. It is also the fundamental building block of the richer scaling limit, the directed landscape. In this talk, I will describe how the Airy sheet is built from asymptotic last passage values along parabolas in the Airy line ensemble via a curious property of the RSK bijection. Based on joint work with Janosch Ortmann and Balint Virag.

Shirshendu Ganguly: Geodesic watermelons in last passage percolation

In last passage percolation on the plane, the k-geodesic watermelon refers to the collection of k oriented paths, disjoint except possibly at the endpoints, that has cumulatively the maximum weight (k-watermelon weight) among all such collections, generalizing the well known k=1 case of the geodesic between a pair of points. For integrable models of last passage percolation, the k-watermelon weights across different values of k admit remarkable bijections to various algebraic objects including the row lengths of Young diagrams and eigenvalues of random matrices, which have proven to be crucial in many recent breakthroughs.

While the geometry of the geodesic has been the object of much study, the k-geodesic watermelon for higher values of k have remained largely unexplored. We will discuss the outcome of a program to investigate such objects, including results establishing sharp exponents governing the fluctuation behavior. The arguments work for any model satisfying a limited set of assumptions about the geodesic weight profile, and rely on a beautiful, deterministic, interlacing property that the geodesic watermelons exhibit.

David Croydon: Invariant measures for KdV and Toda-type discrete integrable systems

This talk is based on joint work with Makiko Sasada (University of Tokyo) and Satoshi Tsujimoto (Kyoto University). I will give a brief introduction to four discrete integrable systems, which are derived from the KdV and Toda lattice equations, and discuss some arguments that are useful in identifying invariant measures for them. As a first key input, I will describe how it is possible to construct global solutions for each of the systems of interest using variants of Pitman's transformation. Secondly, I will present a "detailed balance" criterion for identifying i.i.d.-type invariant measures, and will relate this to approaches used to study various stochastic integrable systems, such as last passage percolation, random polymers, and higher spin vertex models. In many of the examples I discuss, solutions to the detailed balance criterion are given by well-known characterizations of certain standard distributions, including the exponential, geometric, gamma and generalized inverse Gaussian distributions. Our work leads to a number of natural conjectures about the characterization of some other standard distributions.

Sylvie Corteel: Multitype ASEP and Macdonald polynomials

The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice. On the other hand, Macdonald polynomials are a remarkable family of multivariate polynomials which generalize Schur polynomials and Hall-Littlewood polynomials. I'll explain how the study of the multi type ASEP on a ring leads to new formulas for Macdonald polynomials (joint work with Mandelshtam and Williams arXiv:1811.01024). One of the key tools will be the computation of the stationary distribution of the multi-type ASEP in terms of multiline queues due to James Martin (arXiv:1810.10650).

Erik Bates: Endpoints of disjoint geodesics in the directed landscape

Brownian last passage percolation (LPP) has proved a fruitful arena for understanding the geometry of geodesics, or "polymers", arising in models within the Kardar--Parisi--Zhang universality class. The recent construction by Dauvergne, Ortmann, and Virág of a scaling limit for Brownian LPP---the directed landscape---offers a setting in which this understanding can be phrased in terms of fractal geometry. Moreover, the measure theoretic "content" of the directed landscape is carried by exceptional space-time points admitting disjoint geodesics. This talk will quantify, in terms of Hausdorff dimension, the size of some of these exceptional sets, thus shedding light on the landscape's rich fractal structure. (Joint work with Shirshendu Ganguly and Alan Hammond.)

Milind Hegde and Jacob Calvert: The quantitatively Brownian nature of the Airy line ensemble

In these two talks we will discuss a quantitative form of Brownianity enjoyed by the Airy line ensemble, proved using techniques which develop the basic Brownian Gibbs property. The first talk will state the main result, provide some context, and illustrate its applicability with a simple example. The second talk will discuss in broad strokes the framework of the proof and the role of the Brownian Gibbs property, ideas which may be of use in other problems. This is joint work with Alan Hammond

Xuan Wu: Tightness of the KPZ line ensemble

A long standing conjecture in KPZ universality class is the convergence of the solution H_t to KPZ equation to Airy_2 process. In a recent work, Ivan and Hammand proposed a scheme to attack this conjecture through Gibbisan line ensembles, in which H_t is embedded as the top curve. In this talk, we will discuss the tightness of KPZ line ensemble as t varies.

Sourav Sarkar: Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition

In this talk, I will show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion B on every compact interval. In particular, the Airy_1 process is absolutely continuous with respect to B on any compact interval. I will also talk about some recent results here. This is a joint work with Balint Virag.

Lingfu Zhang: Empirical distribution along geodesics in exponential last passage percolation

In the 2D exponential last passage percolation, the geodesic from (0,0) to (n,n) is the up-right path with the maximum sum of the random variables along the path. We study the very local behavior of the geodesic. For every point in the geodesic, we take a constant size box centered around it, and consider the law of the empirical distribution of the boxes along the geodesic. We show that it converges as n grows. The main idea of the proof is to show that the very local behavior around different points in the geodesic are asymptotically i.i.d. Estimates from the exact solvable models are widely used. This is joint work with Allan Sly.