Abstract
For a family of bond percolation models on ℤ2that includes the Fortuin Kasteleyn random cluster model, we consider the family of droplets (i.e., open dual circuits) in anN x Nbox that results, in the percolating regime, from conditioning on the event that the total area inside “large” droplets (diameter at least a large multiple of logN)is at least a given large value A. We show that under such conditioning there is only a single large droplet, and this droplet approximates a Wulff shape. This differs from existing “single droplet” results such as those of Dobrushin, Koteckÿ and Shlosman [10] and Ioffe and Schonmann [15] for the Ising model, in that here the single droplet result must be obtained from surface tension considerations alone, because the minimum total droplet area A is imposed on the system, whereas in the Ising model the possibility of two or more large droplets competes with the possiblity of removing all but the largest of the large droplets and dispersing the contents of the other large droplets in many small droplets.
Research supported by NSF grant DMS-9802368
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Alexander, K. (2002). The Single Droplet Theorem for Random Cluster Models. In: Sidoravicius, V. (eds) In and Out of Equilibrium. Progress in Probability, vol 51. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0063-5_2
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DOI: https://doi.org/10.1007/978-1-4612-0063-5_2
Publisher Name: Birkhäuser, Boston, MA
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