Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom
We usually end for questions at 3:20 PM.
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February 3, 2022, ZOOM: Zhipeng Liu (University of Kansas)
One-point distribution of the geodesic in directed last passage percolation
In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently, but the precise limiting distributions of the geodesic locations remained unknown.
In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.
February 10, 2022, ZOOM: Jacob Calvert (U.C. Berkeley)
Harmonic activation and transport
Models of Laplacian growth, such as diffusion-limited aggregation (DLA), describe interfaces which move in proportion to harmonic measure. I will introduce a model, called harmonic activation and transport (HAT), in which a finite subset of Z^2 is rearranged according to harmonic measure. HAT exhibits a phenomenon called collapse, whereby the diameter of the set is reduced to its logarithm over a number of steps comparable to this logarithm. I will describe how collapse can be used to prove the existence of the stationary distribution of HAT, which is supported on a class of sets viewed up to translation. Lastly, I will discuss the problem of quantifying the least positive harmonic measure as a function of set cardinality, which arises in the study of HAT, and a partial resolution of which rules out predictions about DLA from the physics literature. Based on joint work with Shirshendu Ganguly and Alan Hammond.
February 17, 2022, in person: Pax Kivimae (Northwestern University)
February 24, 2022, ZOOM: Lucas Benigni (University of Chicago)
March 3, 2022, in person: David Keating (UW-Madison)
March 10, 2022, format TBD: Qiang Wu (University of Illinois Urbana-Champaign)
March 24, 2022, in person: Sayan Das (Columbia University)
March 31, 2022, in person: Will Perkins (University of Illinois Chicago)
April 7, 2022, ZOOM: Eliza O'Reilly (Caltech)