4 thoughts on “Geometry and Probability

  1. Gil Kalai Post author

    Dear guest, the first picture is a Peano-like curve (the bold black curve) which (if I remember correctly) is described in terms of a random tree based on Loop arased random walk, and such discrete curves converges in the limit to SLE(8).

    I am not sure about it so regard this answer as a first approximation.

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  2. Gábor Pete

    In that picture, it is more than likely that the white is a Uniform Spanning Tree of the box, i.e., a spanning tree chosen uniformly at random from all possibilities. The red is its dual tree, a Wired Uniform Spanning Tree, a tree if you regard the entire boundary as a single point. They are indeed related to Loop-Erased Random Walk: the path in the tree between any two given points is like a LERW path. Such paths have an SLE(2) scaling limit. And the black Peano curve between red and white indeed has an SLE(8) limit, a space-filling random curve.

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