Discrete integration of the KPZ equation in d>1.

RUBÉN C. BUCETA; MARCOS F. TORRES

Buenos Aires

Congreso; StatPhys 27; 2019

 There is a critical dimension $d_c$  for the Kardar-Parisi-Zhang (KPZ) equation? Simulations of discrete growth models, that belong to the same universality class for the one-dimensional equation, {it e.g.} Restricted Solid-on-Solid, Ballistic Deposition, etc, do not show any sign of its existence. At the same time, mostly of the renormalization group techniques applied find two main results: The first one, a value $d_c$ close to $4$ and the second one, in the same way that for discrete models, a non existent critical dimension. In this work we propose to study the KPZ critical dimension through its discrete integration. As all equations that have a nonlinear term like $(abla h)^2$, the integration of the KPZ equation diverges. This divergence happen when the height difference between two neighboring points in the interface surpass a critical value, which depends of the nonlinear intensity $lambda$. We introduce and characterize  an integration method which consists of directly limiting the value taken by the KPZ nonlinearity, in a way that only affects the divergences while keeping all the properties of the continuous KPZ equation. We use it to show that, at least from the discrete integration of the KPZ equation, $d=4$ is not the critical dimension.