On shell renormalization scheme

The KPZ-equation (named after its creators Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang) is a non-linear stochastic partial differential equation. It describes the temporal change of the height $h(\overrightarrow{x},t)$ at place $\overrightarrow{x}$ and time $t$. It is formally given by

$\frac{\partial h(\overrightarrow{x},t)}{\partial t}=\nu {\mathrm{\nabla }}^{2}h+\frac{\lambda }{2}{\left(\mathrm{\nabla }h\right)}^2+\eta (\overrightarrow{x},t),$

where $\eta (\overrightarrow{x},t)$ is white Gaussian noise with average $⟨\eta (\overrightarrow{x},t)⟩=0$ and second moment $⟨\eta (\overrightarrow{x},t)\eta ({\overrightarrow{x}}^{\prime },t^{\prime })⟩=2D{\delta }^d(\overrightarrow{x}-{\overrightarrow{x}}^{\prime })\delta (t-t^{\prime })$. $\nu$, $\lambda$, and $D$ are parameters of the model and $d$ is the dimension.

By use of renormalization group techniques it has been conjectured that the KPZ equation is the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the SOS model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.

Many models in the field of interacting particle systems, such as the totally asymmetric simple exclusion process, also lie in the KPZ universality class. This class is characterised by models which, in one spatial dimension (1+1 dimension) have a roughness exponent α=1/2, growth exponent β=1/3 and dynamic exponent z=3/2. In order to check if a growth model is within the KPZ class, one can calculate the width of the surface, $W(L,t)$, defined as
$W(L,t)=⟨\frac{1}{L}\underset{0}{\overset{L}{\int }}(h(x,t)-\overline{h}(t){)}^{2}dx{⟩}^{1/2},$
where $\overline{h}(t)$ 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, where we have
$W(L,t)\approx L^{\alpha }f(t/L^z),$
with a scaling function $f(u)$ satisfying
$f(u)\propto \{\begin{array}{lr}{u}^{\beta }& u\ll 1\\ 1& u\gg 1\end{array}$

Sources

• A.-L. Barabási and H.E. Stanley, Fractal concepts in surface growth (Cambridge University Press, 1995)
• Lecture Notes by Jeremy Quastel http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf
• Lecture Notes by Ivan Corwin http://arxiv.org/abs/1106.1596