Physics and mathematics of the Kardar-Parisi-Zhang (KPZ) equation

In 1986, Kardar, Parisi and Zhang introduced a model equation for a growing surface, in the form of a nonlinear partial differential equation with noise[1]. In the original paper they applied a dynamical renormalization group analysis to demonstrate its universal nature, which is one of the first identified non-equilibrium universality classes (KPZ universality class). Since then their equation (KPZ equation) has been accepted as a standard model in non-equilibrium statistical mechanics.

In this talk, we focus on its one dimensional version because it has attracted particular attention in the last decade or so. Mathematically there had been an issue of well-definendness of the equation itself, which was solved by a few different ideas. There is also a high precision experiment using liquid crystal. An important step was the discovery of an exact solution in 2010[2], which confirmed that the height fluctuation is of O(t^(1/3)) and its universal distribution is given by the Tracy-Widom distribution from random matrix theory. Since then there have been a large amount of studies on its generalizations, which now forms a field of “integrable probability”. The activity still continues. Universal behaviors for general initial conditions can now be studied (“KPZ fixed point”). Very recently we have found a direct connection between KPZ systems and free fermion at finite temperature[3].

A remarkable aspect of one dimensional KPZ is its unexpectedly wide universality. For example, KPZ universality is expected to appear in long time behaviors of many one-dimensional Hamiltonian dynamical systems such as anharmonic chains [4]. This is surprising because time-evolution of such systems are deterministic and there are apparently no growing surface with noise. More recently people have observed appearance of KPZ behaviors in dynamical properties of quantum spin chains[5], first in numerical simulations but more recently in real experiments. These discoveries have been attracting considerable attention but theoretical foundations are not yet satisfactory.

In the talk we start from recalling basics of KPZ and then explain these recent developments.

References [1] M. Kardar, G. Parisi, and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56, 889–892 (1986).

[2] T. Sasamoto and H. Spohn, One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett., 104:230602 (2010); G. Amir, I. Corwin, and J. Quastel, Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Comm. Pure Appl. Math., 64, 466– 537 (2011).

[3] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through periodic and free boundary Schur measures, arxiv2204.08420.

[4] H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191–1227 (2014).

[5] M. Ljubotina, M. Znidaric, T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet, Phys. Rev. Lett. 122, 210602 (2019).