Virginia Integrable Probability Summer School 2024

Kailun Chen (Leipzig)
Mallows product measure
q-exchangeable ergodic distributions on the infinite symmetric group were classified by Gnedin-Olshanski (2012). In this paper, we study a specific linear combination of the ergodic measures and call it the Mallows product measure. From a particle system perspective, the Mallows product measure is a reversible stationary blocking measure of the infinite-species ASEP and it is a natural multi-species extension of the Bernoulli product blocking measures of the one-species ASEP. Moreover, the Mallows product measure can be viewed as the universal product blocking measure of interacting particle systems coming from random walks on Hecke algebras. For the random infinite permutation distributed according to the Mallows product measure we have computed the joint distribution of its neighbouring displacements, as well as several other observables. The key feature of the obtained formulas is their remarkably simple product structure. We project these formulas to ASEP with finitely many species, which in particular recovers a recent result of Adams-Balazs-Jay, and also to ASEP(q,M). Our main tools are results of Gnedin-Olshanski about ergodic Mallows measures and shift-invariance symmetries of the stochastic colored six vertex model discovered by Borodin-Gorin-Wheeler and Galashin. This is a joint work with Alexey Bufetov.

Barkat Mian (Mississippi)
On planar Brownian motion singularly tilted through a point potential
We will discuss a special family of two-dimensional diffusions defined over a finite time interval [0, T]. These diffusions have transition density functions that are given by the integral kernels of the semigroup corresponding to the two-dimensional Schr ̈odinger operator with a point potential at the origin. Although, in a few ways, our processes of interest are closely related to two- dimensional Brownian motion, they have a singular drift pointing in the direction of the origin that is strong enough to enable the possibility of visiting there with positive probability. Our main focus is on characterizing a local time process at the origin for these diffusions analogous to that for a one-dimensional Brownian motion.

Samuel McKeown (Wisconsin)
Solvability in a restricted first passage percolation
Strict-weak first passage percolation (SWFPP) is a model of random plane geometry, introduced by Seppäläinen as a simplification of planar FPP for which explicit formulae can be obtained. It is expected to lie in the KPZ universality class. One can apply many of the same techniques which have been used to study last passage percolation, in particular the systematic use of Busemann functions in describing the model's semi-infinite geodesics. SWFPP has the advantage of being solvable for a wider class of distributions and with expressions that tend to be easier to work with. As an application, we show that the tree of semi-infinite geodesics in exponential SWFPP fails to have a natural density near the axis.

We will discuss generative AI (chatGPT, etc.) in research, teaching, and administrative tasks. In the first half of the discussion, Everyone is invited to contribute a short AI success story that others can follow. The second half is an open discussion around AI topics.

Seok Hyun Byun (Clemson)
Dimer bijections, Aztec triangles, and spanning forests
In this talk, we first recall a Temperley’s classical bijection (between spanning trees of a graph and dimer configurations of a related graph) and a related dimer bijection. Then, we present a new result, which extends the aforementioned bijections. As an application, we answer a question posed by Corteel, Huang, and Krattenthaler on finding an explicit bijection between domino tilings of two similar regions. If time permits, we also introduce a new combinatorial object that is equinumerous to domino tilings of Aztec triangles. This talk is based on a joint work with Mihai Ciucu.

Sabrina Gernholt (Bonn)
Fluctuations of a tagged particle in TASEP under influence of a deterministic wall
We consider a totally asymmetric simple exclusion process on Z with step initial condition and with the presence of a rightward-moving wall that prevents the particles from jumping. Our aim is to determine the limiting distribution of a tagged particle whose fluctuations are influenced by the wall around multiple macroscopic times.

Matthew Nicoletti (MIT)
Perfect t-embeddings of the uniform hexagon
We construct and study the asymptotic properties of "perfect t-embeddings" of uniformly weighted hexagon graphs. Hexagon graphs are subgraphs of the honeycomb lattice, and the corresponding dimer model is equivalent to the model of uniformly random lozenge tilings of the hexagon. We provide exact formulas describing the perfect t-embeddings of these graphs, and we use these to prove the convergence of naturally associated discrete surfaces (coming from the "origami maps") to a maximal surface in Minkowski space carrying the conformal structure of the limiting Gaussian free field (GFF). The emergence of such a maximal surface is predicted to hold for a large class of dimer models by Chelkak, Laslier, and Russkikh. In addition, we check all conditions of a theorem of Chelkak, Laslier, and Russkikh which uses perfect t-embeddings to prove convergence of height fluctuations to the GFF, and thus we complete give a new proof, via t-embeddings, of convergence to the GFF. This is based on joint work with Marianna Russkikh and Tomas Berggren.

Sergey Berezin (KU Leuven)
Functional CLT for Mittag-Leffler Ensemble with Hard Wall
The Mittag-Leffler ensemble restricted to an origin-centered disk in the bulk of the droplet features a natural hard edge at the boundary of the disk. This boundary is known as the hard wall. We will discuss a recent functional central limit theorem for dynamical additive statistics in the hard-edge scaling, i.e., at the distance O(1/n) from this hard wall.

Trung Vu (Illinois Urbana-Champaign)
Arctic curves of the T-system with Slanted Initial Data
The octahedron recurrence/equation can be viewed as a 2+1-dimensional discrete evolution equation. Generalizing the study of Di Francesco and Soto-Garrido, we consider initial data along parallel ``slanted" planes perpendicular to an arbitrary admissible direction (r,s,t) in Z_+^3. The solution of the T-system is interpreted as the partition function of a dimer model on some suitable ``pinecone" graph introduced in Bousquet-Mélou, J. Propp, and J. West. The T-system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system.

Xincheng Zhang (Toronto) The totally asymmetric exclusion process and the fluctuations around its shock
The totally asymmetric exclusion process(TASEP) is one of the solvable models in the KPZ universality class. When TASEP starts with certain initial conditions, it presents shocks in the evolution. Particles near the shocks have fluctuations on the t^{1/3} scale. For one type of initial condition, the fluctuations are known to be on the t^{1/2} scale due to the initial randomness. In this talk, I will describe how to see the t^{1/3} fluctuations for these initial conditions.