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Asymptotics for 2D critical and near-critical first-passage percolation

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We study Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability p and 1-p, respectively. At p=1/2, we obtain explicit limit theorems for the point to point passage times a_{0,n} and the passage times between boundary points of the upper half-plane. For the supercritical phase, we give exact asymptotics for the passage times from the origin to the infinite cluster with 0-time sites, as p tending to 1/2. The proof relies on the convergence of the percolation exploration path to SLE , and the collection of interface loops to CLE .

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