KPZ Universality

Introduction

The following introduction is from KPZ universality class. Many interacting particle systems, such as the totally asymmetric simple exclusion process, lie in the KPZ universality class. This class is characterized by the following critical exponents in one spatial dimension (1 + 1 dimension): the roughness exponent $\alpha ={\tfrac {1}{2}}$, growth exponent $\beta ={\tfrac {1}{3}}$, and dynamic exponent $z={\tfrac {3}{2}}$. In order to check if a growth model is within the KPZ class, one can calculate the width of the surface:

$W(L,t)=\left\langle {\frac {1}{L}}\int _{0}^{L}{\big (}h(x,t)-{\bar {h}}(t){\big )}^{2}\,dx\right\rangle ^{1/2},$

where ${\bar {h}}(t)$ is the mean surface height at time $t$ and $L$ is the size of the system. For models within the KPZ class, the main properties of the surface $h(x,t)$ can be characterized by the Family–Vicsek scaling relation of the roughness

$W(L,t)\approx L^{\alpha }f(t/L^{z}),$

with a scaling function $f(u)$ satisfying

$f(u)\propto {\begin{cases}u^{\beta }&\ u\ll 1\\1&\ u\gg 1\end{cases}}$

In 2014, Hairer and Quastel showed that more generally, the following KPZ-like equations lie within the KPZ universality class

${\frac {\partial h({\vec {x}},t)}{\partial t}}=\nu \nabla ^{2}h+P\left(\nabla h\right)+\eta ({\vec {x}},t)\;,$

where $P$ is any even-degree .

A family of processes that are conjectured to be universal limits in the (1+1) KPZ universality class and govern the long time fluctuations are the Airy processes and the KPZ fixed point.

Discrete models

• Asymmetric Simple Exclusion Process(ASEP)

• Totally Asymmetric Simple Exclusion Process(TASEP)

• PushTASEP

Here we go over the types of initial data:

• Types of Initial data