The Kardar-Parisi-Zhang equation and universality class

By Ivan Corwin. 2011

280 citations recorded by the source corpus.

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- a non-linear stochastic partial differential equation -- known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3766458a8991b48be4316c467b6933503c281f98

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Citations in the corpus

Lectures on singular stochastic PDEs (2015, background): “Excellent surveys on the KPZ equation and related areas are [6, 28, 29].”
The nonperturbative functional renormalization group and its applications (2020, background): “dimensions, sustained by a wealth of exact results [417], and by high-precision measurements in turbulent convection of liquid crystals [418, 419].”
Solving the KPZ equation (2011, background): “ larger class of systems, namely the systems in the KPZ universality class which is associated to the dynamic scaling exponents 3 2 , see for example [BQS11]. We refer to the excellent review article [Cor12] for many more references and a more detailed historical account of the KPZ equation. Over th”
Stationary cocycles and Busemann functions for the corner growth model (2015, methodology): “r growth model. Undersomemomentandregularityassumptionsontheweights,thecornergrowth model is expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. (For a review of KPZ universality see [18].) The ﬂuctuations of G0,⌊nξ⌋are expected to have order of magnitude n1/3 and limit distribution”
Emergent statistical mechanics of entanglement in random unitary circuits (2018, background): “χ(t) is a random variable whose magnitude is of order 1 at late times, whose probability distribution is universal and given by the Tracy-Widom distribution F1 [67–80].”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, background): “It is not a review of recent mathematical approaches nor a technical guide to solve the problems, for which the readers are referred to more appropriate reviews [14, 15, 17, 18] and references therein.”