Distribution of sizes of erased loops for loop-erased random walks

We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability P(l) of generating a loop of perimeter l is expressible in terms of the probability P-st(I) of forming a loop of perimeter I when a bond is added to a random spanning tree on the same graph by the simple relation P(l)= P-st(l)/l. On d-dimensional hypercubical lattices, P(l) varies as l(-sigma) for large l, where sigma = 1+2/z for 1<d<4, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with (d) over tilde<2 this relation is modified to sigma=1+2 (d) over tilde/((d) over tilde z), where (d) over tilde is the Hausdorff and (d) over tilde is the spectral dimension of the fractal.