Random walks on critical site percolation clusters converge to CLE6 Brownian motion and intrinsic metrics converge to CLE6 geodesics, implying existence of chemical distance, resistance, and spectral dimension exponents that satisfy Einstein relations.
Blanc-Renaudie, N
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Proves that the giant component in hypercube bond percolation at p = c/d > 1 has diameter Θ(d) and lazy random walk mixing time Θ(d²).
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The scaling limit of random walk and the intrinsic metric on planar critical percolation
Random walks on critical site percolation clusters converge to CLE6 Brownian motion and intrinsic metrics converge to CLE6 geodesics, implying existence of chemical distance, resistance, and spectral dimension exponents that satisfy Einstein relations.
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Diameter and mixing time of the giant component in the percolated hypercube
Proves that the giant component in hypercube bond percolation at p = c/d > 1 has diameter Θ(d) and lazy random walk mixing time Θ(d²).