End-to-End Distance from the Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

In [BEI92] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals ${{\frac{{1}}{{|x|^2}}}}$. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) βc, the Green's 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 ${{\sqrt{{T}}\log^{{\frac{{1}}{{8}}}}T {{\left({{1+O{{\left({{\frac{{\log\log T}}{{\log T}}}}\right)}} }}\right)}}}}$, which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice ℤ4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green's function, and requires detailed properties of the Green's function throughout a sector of the complex β plane. These estimates are derived in a companion paper [BI02].