now a loop-erased random walk starting from v and stopped when hitting the "boundary" of G, i.e. the vertices of G which correspond to the boundary of D. Then the conjectures are As ε goes to zero the distribution of the path converges to some distribution on simple paths from x to the boundary of D (different from Brownian motion, of course — in 2 dimensions paths of Brownian motion are not simple). This distribution (denote it by ) is called the scaling limit of loop-erased random walk. These distributions are conformally invariant. Namely, if φ is a Riemann map between D and a second domain E then The Hausdorff dimension of these paths is 5/4 almost surely. The first attack at these conjectures came from the direction of domino tilings. Taking a spanning tree of G and adding to it its planar dual one gets a domino tiling of a special derived graph (call it H). Each vertex of H corresponds to a vertex, edge or face of G, and the edges of H show which vertex lies on which edge and which edge on which face. It turns out that taking a uniform spanning tree of G leads to a uniformly distributed random domino tiling of H. The number of domino tilings of a graph can be calculated using the determinant of special matrices, which allow to connect it to the discrete Green function which is approximately conformally invariant. These arguments allowed to show that certain measurables of loop-erased random walk are (in the limit) conformally invariant, and that the expected number of vertices in a loop-erased random walk stopped at a circle of radius r is of the order of . In 2002 these conjectures were resolved (positively) using Stochastic Löwner Evolution. Very roughly, it is a stochastic conformally invariant ordinary differential equation which allows to catch the Markov property of loop-erased random walk (and many other probabilistic processes). Three dimensions The scaling limit exists and is invariant under rotations and dilations. If denotes the expected number of vertices in the loop-erased random walk until it gets to a distance of r, then where ε, c and C are some positive numbers (the numbers can, in principle, be calculated from the proofs, but the author did not do it). This suggests that the scaling limit should have Hausdorff dimension between and 5/3 almost surely. Numerical experiments show