Abstract
We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (freeze) as soon as their becomes at least N, for some parameter N ≥ 1. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites). Diameter-frozen and volume-frozen percolation have been studied in previous works ([25, 11] and [27, 26], resp.), and they display radically different behaviors. These works adopt the rule that the of a frozen cluster stays vacant forever, and we investigate the influence of these boundary conditions in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case.
Highlights
1.1 Frozen percolationIn statistical physics, the phenomenon of self-organized criticality refers, roughly speaking, to the spontaneous arising of a critical regime without any fine-tuning of a parameter
The phenomenon of self-organized criticality refers, roughly speaking, to the spontaneous arising of a critical regime without any fine-tuning of a parameter
The critical regime of independent percolation is of particular interest, and arises in models of forest fires [5, 24], displacement of oil by water in a porous medium [28, 4], diffusion fronts [19, 17], and in frozen percolation, the topic of the present paper
Summary
The phenomenon of self-organized criticality (or SOC for short) refers, roughly speaking, to the spontaneous (approximate) arising of a critical regime without any fine-tuning of a parameter. One may ask whether for all applications these “boundary conditions” are always the most natural, and if tweaking them would lead to a different macroscopic behavior This leads us to discuss modified (diameter- and volume-) frozen percolation processes, where, informally speaking, the sites adjacent to a frozen cluster become black (and may freeze) at a later time. V just becomes black (and may become frozen at a later time) These modified processes are denoted by PdNiam and PvNol, and it is not difficult to see that they are well-defined, since they can be seen as finite-range interacting particle systems
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