The scaling limit of the intrinsic metric and simple random walk on 2D critical percolation clusters
Abstract: We show that the CLE(κ) gasket for each κ in (4,8), the range of κ values where the loops can hit each other, themselves, and the domain boundary, can be equipped with: 1) A canonical intrinsic metric and 2) A canonical ”Brownian motion” (a continuous Markov process living in the gasket). We also consider critical percolation on the triangular lattice T and show that: 1) The shortest path distance and 2) The simple random walk on large clusters jointly converge in the scaling limit to our continuum metric and Brownian motion on CLE(6). Based on joint works with Valeria Ambrosio, Irina Dankovic, Maarten Markering, and Yizheng Yuan.
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