End-to-end Distance from the Greens Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Greens function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) beta^c, the Greens function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Greens function, and requires detailed properties of the Greens function throughout a sector of the complex beta plane. These estimates are derived in a companion paper [math-ph/0205028].