Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

Abstract

This is the second of two papers on the end-to-end distance of a weakly self-repelling walk on a four dimensional hierarchical lattice. It completes the proof that the 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. - Apart from completing the program in the first paper, the main result is that the Green's function is almost equal to the Green's function for the Markov process with no self-repulsion, but at a different value of the killing rate βwhich can be accurately calculated when the interaction is small. Furthermore, the Green's function is analytic in βin a sector in the complex plane with opening angle greater than π.