Abstract
We study atypical behavior in bootstrap percolation on the Erdős–Rényi random graph. Initially a set S is infected. Other vertices are infected once at least r of their neighbors become infected. Janson et al. (Ann Appl Probab 22(5):1989–2047, 2012) locates the critical size of S, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this work, we calculate the rate function for the event that a small set S eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.
Highlights
Bootstrap percolation was originally proposed by physicists [12,29] to model the phase transition observed in disordered magnets
We focus on this special case, we think our methods could be useful in studying the large deviations of any Markovian growth or Communicated by Giulio Biroli
Of particular interest is the critical size at which point a uniformly random initial set S0 is likely to infect most of the graph
Summary
Bootstrap percolation was originally proposed by physicists [12,29] to model the phase transition observed in disordered magnets. Since a large literature has developed, motivated by beautiful results, e.g. [8,10,22,31], and a variety of applications across many fields, see e.g. [1,2] and references therein. We consider the spread of an infection by the r -neighbor bootstrap percolation dynamics on the Erdos–Rényi [15] graph Gn,p, in which any two vertices in [n] are neighbors independently with probability p. We focus on this special case, we think our methods could be useful in studying the large deviations of any Markovian growth or Communicated by Giulio Biroli
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