Probability and Stochastic Analysis - University of Bonn

Animation of an anisotropic particle system in the AKPZ class of growth models in 2+1 dimensions

This is a visualization of the model we studied in our paper Anisotropic growth of random surfaces in 2+1 dimensions.

  • Configurations.The circles represents the positions of particles in (x,n)-coordinates (n is called the level of the particle). The particles positions at time t are denoted by (xkn(t),n), for k=1,...,n, and n=1,..., number of levels. The particles satisfy at any time the interlacing condition, xkn+1(t) < xkn(t) <= x k+1n+1(t).
  • Initial condition. At time t=0 we have xkn(0)=k-n, and it corresponds to the packed system of particles on the screen.
  • Evolution: Each particle has one clock, which all independently ring at rate 1. When the clock of a particle rings, it tries to jump to the right. Whether the jump occurs, is determined only by the particles' configuration below it, namely, the jump occurs only if the interlacing condition on the levels below is still satisfied. It can happen that the interlacing condition above the particle that jumped would not be satisfied anymore. In that case, the minimal amount of particles which have to be moved to keep the interlacing structure to hold are moved by one unit too (we call it pushing).


2+1 dimensional particle system
Angle θ
Angle φ
Visualization options
For particles
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