Editing Loop-erased random walk (section)

===Two dimensions===
In two dimensions, arguments from [[two-dimensional conformal field theory|conformal field theory]] and simulation results led to a number of exciting conjectures. Assume ''D'' is some [[simply connected]] [[Domain (mathematical analysis)|domain]] in the plane and ''x'' is a point in ''D''. Take the graph ''G'' to be

:<math>G:=D\cap \varepsilon \mathbb{Z}^2,</math>

that is, a grid of side length ε restricted to ''D''. Let ''v'' be the vertex of ''G'' closest to ''x''. Examine 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&nbsp;— in 2 dimensions paths of Brownian motion are not simple). This distribution (denote it by <math>S_{D,x}</math>) is called the '''scaling limit''' of loop-erased random walk.
* These distributions are [[Conformal map|conformally invariant]]. Namely, if φ is a [[Riemann mapping theorem|Riemann map]] between ''D'' and a second domain ''E'' then

:<math>\phi(S_{D,x})=S_{E,\phi(x)}.\,</math>

*The [[Hausdorff dimension]] of these paths is 5/4 [[almost surely]].

The first attack at these conjectures came from the direction of '''[[domino tiling]]s'''. Taking a spanning tree of ''G'' and adding to it its [[Planar graph|planar dual]] one gets a [[Dominoes|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's function|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 value|expected]] number of vertices in a loop-erased random walk stopped at a circle of radius ''r'' is of the order of <math>r^{5/4}</math>.<ref>{{harvtxt|Kenyon|2000a}}</ref>

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).