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Scaling limit of the largest clusters in critical percolation and FK-Ising models in two dimensions

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Consider the percolation model with parameter p on hexagonal lattice with mesh-size \eta: We paint each hexagon by red with probability p, and blue with probability 1-p, independently from each other. At p = 1/2, macroscopic red and blue clusters (connected components) coexist. We show that at p = 1/2, as \eta tends to 0, the macroscopic red clusters seen as closed subsets of the plane converge to a collection of continuum clusters. Similar result holds for the normalized counting measure of the hexagons in red clusters. We discuss the background of the results above, and their applications including a construction of scaling limit of frozen percolation and alternative constructions for near-critical percolation and for the magnetization field in the critical Ising model. Our results are based on papers by Garban, Pete and Schramm, and it is joint work With Federico Camia and Rene Conijn.

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