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

TL;DR

This paper proves that the expected end-to-end distance of a weakly self-repelling walk in four dimensions scales as rac{rac{1}{2}}{ ext{log}^{1/8}T} and shows the Green's function closely resembles that of a Markov process without self-repulsion, with precise calculations for small interactions.

Contribution

It completes the analysis of the self-avoiding walk in four dimensions by establishing the asymptotic growth law and characterizing the Green's function's relation to a non-interacting process.

Findings

Expected end-to-end distance grows as rac{rac{1}{2}}{ ext{log}^{1/8}T}

Green's function approximates that of a Markov process with adjusted killing rate

Green's function is analytic in eta in a broad complex sector

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 \beta which can be accurately calculated when the interaction is small. Furthermore, the Green's function is analytic in \beta in a sector in the complex plane with opening angle greater than \pi.