Oded Schramm: From Circle Packing to SLE

Values of Brownian intersection exponents, I: Half-plane exponents

“…Remarks. It will be shown [40] that for all κ ≥ 0 the hull K t of SLE κ is a.s. generated by a path. More precisely, a.s. the map γ(t) := g −1 t (W κ t ) is a well defined continuous path in H and for every t ≥ 0 the domain D t is the unbounded connected component of H \ γ([0, t]).…”

On limits of graphs sphere packed in Euclidean space and applications

“…This beautiful result has a long history, we refer to [20] and [15] for further information. When d = 3, very little is known.…”

“…A graph is sphere packable in R d if and only if it is the tangency graph of a collection of d-dimensional balls with disjoint interiors: the balls of the packing correspond to the vertices of the graph and the edges to tangent balls, see Section 2.1. The theory of circle packings of planar graphs is well developed and its relation to conformal geometry is well established, see the beautiful survey [15]. The higher dimensional version is not as neat.…”

Planar Brownian Motion and Complex Analysis

“…The literature on SLE is too vast for me to survey, but an interested reader can find good starting points in [72,73,74].…”