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Two-dimensional critical percolation: the full scaling limit. (English) Zbl 1117.60086
The authors study the Bernoulli site percolation model on the two-dimensional triangular lattice at the threshold \(p_c=1/2\). They consider the set of all interfaces between occupied and non-occupied sites. It is shown that the scaling limit of this set (in the sense of M. Aizenman and A. Burchard [Duke Math. J. 99, No. 3, 419–453 (1999; Zbl 0944.60022)]) is a process of continuum non-simple loops in the plane which is constructed by the authors using chordal SLE\(_6\) paths. It is further proved that this process is conformally invariant and it consists of countably many non-crossing continuous loops that touch each other many times. Moreover, any deterministic point in the plane is surrounded by an infinite family of nested loops with diameters going both to zero and infinity. The proofs are based on the fact that the percolation exploration path converges in distribution to the trace of chordal SLE\(_6\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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