Editing Loop-erased random walk (section)

===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.