The Kardar-Parisi-Zhang (KPZ) fixed point is a universal space-time random field of governing models in the KPZ universality class. We consider the asymptomatic properties of the KPZ fixed point when the height at a specific location becomes large. With the narrow-wedge initial condition, we obtained an explicit formula for distribution of the field after an appropriate scaling. Moreover, there is a phase transition for the conditional field near the given point. I will explain how the conditional field arises in the setting of the directed landscape, which is another central object in the KPZ universality. This is based on joint work with Zhipeng Liu, Ron Nissim and Yizao Wang.

We investigate a model of coexisting populations undergoing genetic drift, mutation and selection. Each population evolves as a measure-valued Fleming-Viot process whose fitness of types is a linear combination of the frequencies of types in the other populations, where the coefficients of this linear combination fluctuate over time. We introduce a family of dual processes which are used to analyze the behavior of the model. We also briefly discuss how an extended version of these duals can be applied to study some related models such as the Fleming-Viot model with frequency-dependent selection and the spatial Lambda-Fleming-Viot process in random environment. This talk is based on joint work with Don Dawson.

The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. In this talk, we will discuss a functional large deviation principle (LDP) for the entire random metric. Applying the contraction principle, our result yields an LDP for the geodesics in the directed landscape. If time permits, we will also mention certain interesting features of the rate function for the geodesic LDP. Based on a joint work with Duncan Dauvergne and Balint Virag.

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