v to w; the distribution of this path is identical to that of a loop-erased random walk from v to w. An alternative view is that the distribution of a loop-erased random walk conditioned to start in some path β is identical to the loop-erasure of a random walk conditioned not to hit β. This property is often referred to as the Markov property of loop-erased random walk (though the relation to the usual Markov property is somewhat vague). It is important to notice that while the proof of the equivalence is quite easy, models which involve dynamically changing harmonic functions or measures are typically extremely difficult to analyze. Practically nothing is known about the p-Laplacian walk or diffusion-limited aggregation. Another somewhat related model is the harmonic explorer. Finally there is another link that should be mentioned: Kirchhoff's theorem relates the number of spanning trees of a graph G to the eigenvalues of the discrete Laplacian. See spanning tree for details. Grids Let d be the dimension, which we will assume to be at least 2. Examine Zd i.e. all the points with integer . This is an infinite graph with degree 2d 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 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 conformal field theory and simulation results led to a number of exciting conjectures. Assume D is some simply connected domain in the plane and x is a point in D. Take the graph G to be that is, a grid of side length ε restricted to D. Let v be the vertex of G closest to x. Examine