Weak universality of interacting particle systems
- In this dissertation we consider random growth phenomenon related to the Kardar-Parisi-Zhang (KPZ) equation. We focus particularly on the weak universality of interacting particle systems. That is, under suitable scalings, a certain class of one-dimensional, weakly irreversible interacting particle systems converge to the KPZ equation. Our discussion emphasizes on both the inter- and intro-model universality. For the former, we derive the exact microscopic Hopf--Cole transformation for the 4-parameter family of Higher Spin Exclusion Processes (HSEPs) introduced by Corwin and Petrov (2016). This is done by exploiting the close relationship between Hopf-Cole transformation and duality. As the HSEPs sit above most of the known integrable models in the KPZ class, we thus obtain the exact microscopic Hopf-Cole transformation for all lower-level models. To demonstrate the weak universality, we further consider a particular weak scaling of the HSEPs, and show the convergence to the KPZ equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated. As for intro-model universality, we analyze a class of non-nearest-neighbor exclusion processes and the corresponding growth models. Our approach is to exploit an approximate Hopf-Cole transformation, to which end we identify the main nonlinearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the KPZ equation.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Mathematics.
|Statement of responsibility
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Li-cheng Tsai
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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