The large connectivity limit of bootstrap percolation

TL;DR

This paper studies bootstrap percolation on large regular random graphs, revealing that phase transition behaviors persist in the large connectivity limit and are influenced by the ratio of threshold to connectivity, with complex bifurcations emerging when this ratio varies randomly.

Contribution

It extends the understanding of bootstrap percolation by analyzing its phase behavior in the large connectivity limit and introduces the effects of random threshold ratios on phase transitions.

Findings

Mixed phase behavior persists at large connectivity.

Multiple phase transitions and bifurcations occur when the threshold ratio is random.

The phase behavior depends critically on the ratio of threshold to connectivity.

Abstract

Bootstrap percolation provides an emblematic instance of phase behavior characterised by an abrupt transition with diverging critical fluctuations. This unusual hybrid situation generally occurs in particle systems in which the occupation probability of a site depends on the state of its neighbours through a certain threshold parameter. In this paper we investigate the phase behavior of the bootstrap percolation on the regular random graph in the limit in which the threshold parameter and lattice connectivity become both increasingly large while their ratio $\alpha$ is held constant. We find that the mixed phase behavior is preserved in this limit, and that multiple transitions and higher-order bifurcation singularities occur when $\alpha$ becomes a random variable.