Course 7 Bootstrap and jamming percolation

Abstract

The chapter reviews the results for Bootstrap Percolation (BP), and the more recently introduced Jamming Percolation (JP) models. BP is a cellular automaton with a discrete time deterministic dynamics in which, at each time unit the configuration updates according to a local and translation invariant rule: an empty site remains empty and a filled one is emptied if, and only if, it has at least “m” empty nearest neighbors. BP is successfully used to model other systems, both in physics and in different fields such as biology (infection models), geology (crack formation models). and more recently in computer science. BP is regarded as the functional units of memory arrays or computer networks and the study of BP on the corresponding graphs is relevant to analyze stability against random damage, namely to find the minimal value of connectivity which is necessary to maintain a proper level of inner communication and data mirroring. JP models are introduced with the aim of finding a jamming transition, namely a first-order critical transition on a finite dimensional lattice. In this chapter, the easiest example of a JP model is reviewed—namely, the two-dimensional model, which are introduced and dubbed Spiral Model (SM).