In 2-dimensional critical percolation, with positive probability, there is a path that connects the left and right side of a square box. The chemical distance is the expected length of the shortest such path conditional on its existence. In this talk, I will introduce the best known estimates for chemical distance. I will then discuss analogous estimates for the radial chemical distance (the expected length of the shortest path from the origin to distance n), as well as on-going work to extend these estimates to the random cluster model. A portion of this talk is based on joint work with Philippe Sosoe.
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Chemical distance for 2d critical percolation and random cluster model