Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

Details Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

2011-12-09

arxiv.org/abs/1112.2017v2

Mathematics

Probability

40 pages, 14 figures v2: Section 8 has been revised. Other minor corrections and clarifications

Scientific paper

In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. However our proof uses a series of transformations that allow us to apply the convergence in the site percolation case on the hexagonal lattice to obtain certain estimates that is enough for us to prove the convergence in the case of bond percolation on the square lattice.

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