Random spatial growth with paralyzing obstacles

TL;DR

This paper models spatial growth with paralyzing obstacles on graphs, analyzing conditions for well-defined growth and the distribution of cluster sizes, revealing exponential tail behavior and connections to invasion percolation.

Contribution

It introduces a novel growth model with red obstacles, establishes conditions for its well-definedness on infinite graphs, and links invasion percolation to critical Bernoulli percolation.

Findings

Model is well-defined with positive red density and small white density.

Green cluster sizes have exponential tail distribution.

Established a new relation between invasion percolation and critical Bernoulli percolation.

Abstract

We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no `inter-green' competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as red substance itself. In our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation. The main issues we investigate are whether the model is well-defined on an infinite graph (e.g. the $d$-dimensional cubic lattice), and what can be said about the distribution of the size of a green cluster just before it is paralyzed. We show that, if the initial density of red vertices is positive, and that of white vertices is sufficiently small, the model is indeed well-defined and the above distribution has an exponential tail. In fact, we believe this to be true whenever the initial density of red is positive. This research also led to a relation between invasion percolation and critical Bernoulli percolation which seems to be of independent interest.