Late points of random walk
This page presents some simulations for the late points of a random walk on the three-dimensional torus of side length 400, which shows the phase transition from a phase with only isolated and independent points, to a phase where points can be at small distance from one another. We refer to the article Phase transition for the late points of random walk for detail. The mathematica notebook to run this simulation is available here, and you are welcome to use and modify it.Â
Late points
The simulation displays the late points of the random walk at level α. The points in red are not isolated (i.e. they have at least one neighbor in the set of late points), whereas the points in blue are isolated. The distance between the two closest vertices in the set of late points is also computed. In dimension three, the phase transition occurs at α∗  ≈ 0.67.
Independent Bernoulli
In comparison, independent Bernoulli vertices with the same density have only isolated points for all α>1/2, as shown by the following simulations.
