Editing Loop-erased random walk (section)

==Grids==
Let ''d'' be the dimension, which we will assume to be at least 2. Examine '''Z'''<sup>''d''</sup> i.e. all the points <math>(a_1,...,a_d)</math> with integer <math>a_i</math>. This is an infinite graph with degree 2''d'' when you connect each point to its nearest neighbors. From now on we will consider loop-erased random walk on this graph or its subgraphs.

===High dimensions===
The easiest case to analyze is dimension 5 and above. In this case it turns out that there the intersections are only local. A calculation shows that if one takes a random walk of length ''n'', its loop-erasure has length of the same order of magnitude, i.e. ''n''. Scaling accordingly, it turns out that loop-erased random walk converges (in an appropriate sense) to [[Brownian motion]] as ''n'' goes to infinity. Dimension 4 is more complicated, but the general picture is still true. It turns out that the loop-erasure of a random walk of length ''n'' has approximately <math>n/\log^{1/3}n</math> vertices, but again, after scaling (that takes into account the logarithmic factor) the loop-erased walk converges to Brownian motion.

===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).

===Three dimensions===
The scaling limit exists and is invariant under rotations and dilations.<ref>{{harvtxt|Kozma|2007}}</ref> If <math>L(r)</math> denotes the expected number of vertices in the loop-erased random walk until it gets to a distance of ''r'', then

:<math>cr^{1+\varepsilon}\leq L(r)\leq Cr^{5/3}\,</math>

where ε, ''c'' and ''C'' are some positive numbers<ref>{{harvtxt|Lawler|1999}}</ref> (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 <math>1+\varepsilon</math> and 5/3 almost surely. Numerical experiments show that it should be <math>1.62400\pm 0.00005</math>.<ref>{{harvtxt|Wilson|2010}}</ref>

{{more footnotes needed|date=June 2011}}