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

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

- Event time: 1:00pm until 2:00pm
- Event date: 19th January 2024
- Speaker: Professor Tomohiro Sasamoto (Tokyo Institute of Technology)
- Location: Higgs Centre Seminar Room, Room 4305, James Clerk Maxwell Building (JCMB) (James Clerk Maxwell Building (JCMB)) James Clerk Maxwell Building Peter Guthrie Tait Road Edinburgh EH9 3FD GB

### Event details

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).

#### Event resources

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