The Kardar-Parisi-Zhang equation and universality class (1106.1596v4)

Abstract: 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.

• The paper provides a comprehensive survey of two decades of developments surrounding the Kardar-Parisi-Zhang (KPZ) equation and its associated universality class.
• A key advancement highlighted is the derivation of an exact formula for one-point distributions of the KPZ equation, connecting it to the Tracy-Widom distribution from random matrix theory.
• The KPZ universality class is shown to be relevant beyond theory, linking to physical systems like the WASEP, directed polymers in random media, and experimental phenomena such as bacterial growth and surface deposition.

The Kardar-Parisi-Zhang Equation and Universality Class

The paper by Ivan Corwin provides an extensive survey of the developments surrounding the Kardar-Parisi-Zhang (KPZ) equation and its associated universality class, consolidating progress made over the last two and a half decades in both theoretical and applied aspects. The KPZ equation is a nonlinear stochastic partial differential equation (SPDE) which serves as a fundamental model for the dynamics of growing interfaces and is widely recognized for its role in describing a new universality class of stochastic processes.

Overview of Universality Classes

In statistical physics and probability theory, universality classes categorize the behavior of systems that share critical exponents and scaling functions, regardless of microscopic specifics. Historically, Gaussian universality has dominated, describing systems whose random fluctuations conform to Gaussian distributions. However, the KPZ universality class emerges as a distinct entity characterized by peculiar scalings, notably a 1/3 power-law for fluctuations, and non-Gaussian statistics.

Main Contributions

The investigation into KPZ universality spans three significant areas:

1. Weakly Asymmetric Simple Exclusion Process (WASEP): This involves approximations where the KPZ equation serves as the scaling limit for asymmetric transport processes. The classical simple exclusion process provides a discrete model, revealing the macroscopic KPZ dynamics upon suitable scaling.
2. Exact Solution of KPZ Equation: The paper highlights a milestone achievement—the derivation of an explicit formula for one-point distributions of the KPZ equation under narrow wedge initial conditions. This formula converges to the Tracy-Widom distribution in certain limits, reinforcing KPZ's connection with random matrix theory.
3. Directed Polymers in Random Media: The KPZ class connects to a broad family of systems through its analysis of polymers directed in a disordered environment, representing a valuable model for understanding disorder and randomness in physical settings.

Technical Innovations and Results

The paper explores the precise construction of the KPZ equation's Hopf-Cole solution, detailing the mathematical rigor required to transform ill-posed nonlinearities into solvable forms using exact formulas. A notable contribution is the confirmation of the KPZ universality through solving discrete models (such as ASEP and related growth models), which converges to the continuous setting upon taking a limit.

Key results include:

• Scaling Analysis: Crucial evidence shows KPZ scaling exponents do not emerge arbitrarily but in intriguingly specific distributions aligned with Timothy Widom and Baik-Deift-Johansson's studies of random matrices.
• Exact Formulas: Applications of techniques from combinatorics and representation theory furnish the exact one-point distribution formula for the KPZ equation's solutions, offering new insights into stochastic growth models.

Implications and Future Directions

Corwin’s synthesis underscores the broad applicability of the KPZ universality class beyond mere theoretical interest, extending into experimental realms including bacterial growth, turbulent fluid dynamics, and surface deposition processes. Furthermore, the paper speculates on several open problems crucial for advancing the field:

• Universality Extensions: It emphasizes the necessity for extending solvability and universality results to various initial conditions and beyond the current dimensional restrictions.
• Multi-point and Temporal Statistics: Enhancing understanding of spatial correlations and dynamics in the KPZ regime.
• Rigorous Connections: Establishing rigorous connections between KPZ theory and quantum integrable systems.

Conclusion

Corwin's survey is a comprehensive exposition of the KPZ universality, emphasizing both the theoretical underpinnings and the profound implications of the KPZ equation in diverse physical contexts. The insights gleaned from exact solutions of the KPZ equation not only verify its pivotal role in nonlinear stochastic dynamics but also pave a path toward resolving outstanding concepts in statistical physics. Future research, guided by the outlined challenges, will likely continue to reveal deeper connections between the KPZ universality class and other sectors of mathematical and physical sciences.