Schedule for: 20w5203 - Permutations and Probability (Online)

A brief introduction video from BIRS staff with important logistical information

Assign district cards to the vertices of a finite, connected graph G(V, E) with |V|=n. The interchange process is a card shuffling scheme generated by flipping the cards at the ends of the edges of the graph. The exclusion process follows the same dynamics where instead each vertex is assigned one of two colors. In this talk, I will discuss recent developments concerning the mixing time of these two processes on the interval of length n. I will focus in particular on the asymmetric exclusion process with open boundaries, where particles are allowed to jump in and out from the ends of the interval. This is joint work with Gantert and Schmid.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

I will describe a framework for proving convergence to the directed landscape, the central limit object in the KPZ universality class. The directed landscape is a random scale-invariant `directed' metric on the plane. One highlight of this work is that the scaling limit of the longest increasing subsequence in a uniformly random permutation is a geodesic in the directed landscape. Joint work with Balint Virag.

Joint work with Jacopo Borga, https://arxiv.org/abs/2008.09086 . The theory of permutons allows us to express scaling limits of the diagram of permutations. Scaling limits of uniform elements in various classes of pattern-avoiding permutations have attracted a fair amount of attention lately. We show such a result for Baxter permutations, a famous class of permutations avoiding generalized patterns.

The oriented swap process is a model for a random sorting network, in which N particles labeled 1,...,N arranged on the discrete lattice [1,N] start out in increasing order and then perform successive adjacent swaps at random times until they reach the reverse configuration N,...,1. An open problem from 2009 asked for the limiting law of the absorbing time of the process. In recent joint works with Bisi-Cunden-Gibbons and Bufetov-Gorin, we resolved this problem by showing that the limiting law is the Tracy-Widom GOE distribution, aka F_1. I will tell the story of this result and how it came to be discovered and proved, which involves connections to recent works by Borodin-Gorin-Wheeler, Dauvergne and Galashin, and a new family of distributional identities relating the behavior of the oriented swap process in a surprising way to last passage percolation. Some of these identities are still conjectural and hint at the existence of symmetries in the oriented swap process, multi-type TASEP and related processes that are still not understood. As a side treat for algebraic combinatorics enthusiasts, the RSK, Burge and Edelman-Greene maps will also make an appearance.