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The KPZ Universality Class and Related Topics
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abstract
These notes are based on a talk given at the 2018 Arizona School of Analysis and Mathematical Physics. We give a comprehensive introduction to the KPZ universality class, a conjectured class of stochastic process with local interactions related to random growth processes in $1+1$ dimensions. We describe some of the characteristic properties of the KPZ universality class such as scaling exponents and limiting statistics. In particular, we aim to extract the characteristic properties of the KPZ universality class by understanding the KPZ stochastic partial differential equation by a special discrete approximation given by the asymmetric simple exclusion process (ASEP). The connection with the ASEP is very important as the process enjoys a rich integrability structure that leads to many exact formulas.
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Riemann surfaces for KPZ with periodic boundaries
Known exact finite-volume KPZ fluctuation probabilities are expressed as traces on Riemann surfaces for half-integer polylogarithms, and prior formulas by Prolhac and by Baik and Liu are proved equivalent.
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