Abstract: In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties. We will classify all the ergodic components, establish the ``one force---one solution'' principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic results have their counterparts in the geodesic/polymer language. In the second part, we will present a random point field model that is motivated by the coalescing and monotone structure of the optimal paths in random environments that arise in many KPZ models. The 2/3 transversal exponent from the KPZ scaling becomes a natural parameter for the renormalization action in this model, and can be potentially extended to values other than 2/3. Some preliminary results are given.
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data. Joint work with Shirshendu Ganguly.
Abstract: I'll discuss recent work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.
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