Transitive closure in a polluted environment

TL;DR

This paper introduces a new percolation model based on transitivity in a polluted environment, analyzing phase transitions in the spread of occupied edges under various conditions and identifying critical probabilities for different regimes.

Contribution

The paper develops a novel percolation model inspired by existing models, analyzing phase transitions and identifying critical probabilities in both linear and bounded degree graphs.

Findings

Identified three regimes of edge occupation in linear graphs based on open edge probabilities.

Determined the critical probability threshold for sparse versus full occupation in bounded degree graphs.

Provided conjectures and open problems for future research in percolation theory.

Abstract

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph $G_0$ of initially occupied edges on $n$ vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When $G_0$ is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where $G_0$ is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is $(\log n)^{-1/2+o(1)}$. We conclude with several conjectures and open problems.