The growth exponent for planar loop-erased random walk.

Let Gr(n) be the expected number of steps of a planar loop-erased random walk from the origin to the circle of radius n. It was proved by Richard Kenyon that Gr(n) is logarithmically asymptotic to n raised to the 5/4 power. His proof uses domino tilings to compute asymptotics for the number of uniform spanning trees of rectilinear regions of the plane, and is specific to simple random walk on the integer lattice. In this paper we give a new proof that the growth exponent for loop-erased random walk is 5/4, valid for irreducible bounded symmetric random walks on discrete lattices of the plane. Our proof uses the convergence of loop-erased random walk to Schramm-Loewner evolution with parameter 2. We use an intersection exponent for the latter to deduce that the probability that a loop-erased random walk and an independent random walk do not intersect up to leaving the ball of radius n is logarithmically asymptotic to n to the -3/4 power. We then relate the latter probability to Gr(n).