The Kardar-Parisi-Zhang equation in $d\geq 3$ and the Gaussian free field

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a singular stochastic partial differential equation (SPDE) and belongs to a large class of models known as the KPZ universality class, which is believed to exhibit very different behavior than Gaussian universality class and describe the long-time of a wide class of systems including some noisy SPDEs, driven lattice gases, randomly growing interfaces and directed polymers in random media. In spatial dimension one, recently this class has been studied extensively based on approximations by exactly solvable models. which no longer exist if perturbations appear in the approximating models, or when higher dimensional models are investigated. When the spatial dimension is at least three, it was conjectured that two disjoint universality classes co-exist when the long-time/large-scale behavior of the solutions are studied. We will report some recent progress along these directions.

mathematical physicsanalysis of PDEsprobability

Audience: researchers in the topic

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