September 26: Paul Duncan (Indiana)

Title: Lattice Gauge Theory via a Cellular Representation

Abstract: A central question in the Potts model of magnetism concerns the relationships between distant spins at different temperatures. These correlations were fruitfully studied using the random-cluster model of percolation, which encodes the spin correlations in a dependent random graph. We will discuss an extension of this idea to higher dimensions, in which Potts lattice gauge theory, a model which assigns spins to edges instead of vertices, can be understood through a dependent random cubical complex. Based on joint work with Ben Schweinhart.

October 10: Fall Break, No Seminar.

October 24: Matthew Lerner-Brecher (MIT)

Title: Edge Limits of Random Matrix Ensembles at Zero Temperature

Abstract: The Laguerre beta ensemble, one of the three classical random matrix ensembles, models the behavior of eigenvalues of random covariance matrices. The beta parameter here not only distinguishes between the real, complex, and quaternionic cases (beta = 1,2,4 respectively), but also functions as the inverse temperature when the ensemble is viewed through the lens of interacting particle systems.

In this talk, we will discuss a multilevel extension of the Laguerre beta ensemble at zero temperature (i.e. infinite beta) with emphasis on its hard-edge behavior. In particular, we will see that, as the number of eigenvalues goes to infinity, this behavior can be captured by a discrete time Gaussian process with deep connections to Bessel functions. Our work here builds off Gorin-Kleptsyn (2020), which studied the soft edge behavior of the corresponding Gaussian case, and we will spend some time expounding on interesting connections that arose between our results and theirs.

October 31: Mikhail Tikhonov (Virginia)

Title:  Noncolliding q-exchangeable random walks

Abstract: This talk is about a system of noncolliding q-exchangeable random walks on the set of nonnegative integers making steps straight or down from some initial configuration. Following Gnedin and Olshanski, we say that a (single) simple random walk is called a q-exchangeable random walk if, under an elementary transposition of the neighboring steps, the probability of the trajectory is multiplied by a parameter q in the unit interval [0, 1]. Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via Doob’s h-transform. We show that the trajectory forms a determinantal point process and obtain its kernel, limit shape, and local statistics. The work is based on joint research (arXiv:2303.02380) with L. Petrov.

November 7: Jacob Richey (Rényi Institute)

Title: Stochastic abelian particle systems and self-organized criticality

Abstract: Activated random walk (ARW) is an ‘abelian’ particle system that conjecturally exhibits complex behaviors which were first described by physicists in the 1990s, namely self organized criticality and hyperuniformity. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.

 

November 14: Zongrui Yang (Columbia)

Title: Stationary measures for integrable models on a strip

Abstract: We prove that the stationary measures for the geometric last passage percolation (LPP) and log-gamma polymer models on a diagonal strip are given by the marginal distributions of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we obtain the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.

November 21:

November 28: Thanksgiving Break, No Seminar.

December 5: Jingheng Wang (Ohio State)

Title: TBD

Abstract: TBD