Aldous (Math Proc Camb Philos Soc 128:465–477, 2000) introduced a modification of the bond percolation process on the binary tree where clusters stop growing (freeze) as soon as they become infinite. We investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach L∞ diameter at least N for some parameter N. We show, informally speaking, that in the limit N → ∞, the clusters only freeze in the critical window of site percolation on the triangular lattice. Hence the fraction of vertices that eventually (i. e. at time 1) are in a frozen cluster tends to 0 as N goes to infinity. We also show that the diameter of the open cluster at time 1 of a given vertex is, with high probability, smaller than N but of order N. This shows that the process on the triangular lattice has a behaviour quite different from Aldous’ process. We also indicate which modifications have to be made to adapt the proofs to the case of the N-parameter frozen bond percolation process on the square lattice. This extends our results to the square lattice, and answers the questions posed by van den Berg et al. (Random Struct Algorithms 40:220–226, 2012).
Additional Metadata
Keywords Frozen cluster · Critical percolation · Near critical percolation · Correlation length
THEME Life Sciences (theme 5), Logistics (theme 3)
Publisher Springer Berlin / Heidelberg
Journal Probability Theory and Related Fields
Citation
Kiss, D. (2015). Frozen percolation in two dimensions. Probability Theory and Related Fields, 163, 713–768.