Periodic Kardar-Parisi-Zhang (KPZ) Universality

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Many random growth models, such as fire propagation or bacterial colony growth are believed to share certain universal pattern. Analyzing the mathematical mechanism of such pattern has been an active research area in the last twenty years. Due to the breakthrough progress in the probability and mathematical physics community, an increasing number of models have been successfully analyzed and they are found to share the same large time limiting behaviors. These models are called to belong to the Kardar-Parisi-Zhang (KPZ) universality class, named after Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang who introduced a non-linear stochastic partial differential equation, the so-called KPZ equation, to describe the random growing interfaces. This project aims to study models in the KPZ universality class with spatial periodicity. One goal of this project is to analyze the periodic solvable growth models and understand their universal limiting behaviors under different parameter scales by probing the structure and asymptotics of the Bethe roots associated with these models. The other goal is to develop a new direction to approach the KPZ universality class in the infinite space by obtaining exact results of periodic solvable growth models when the period becomes sufficiently large.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.