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

TL;DR

This paper analyzes the end-to-end distance of a hierarchical self-avoiding walk in four dimensions, showing it grows like rac{rac{T}{}log^{1/8}T, matching conjectures for walks on Z^4.

Contribution

It provides a detailed analysis of the end-to-end distance for a hierarchical self-avoiding walk, confirming conjectured growth laws in four dimensions.

Findings

Expected distance grows as  \,  \,  log^{1/8}T

Uses inverse Laplace transforms and complex Green's function analysis

Matches conjectured behavior for self-avoiding walks in 4D

Abstract

In [BEI] 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 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 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^{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 Green's function, and requires detailed properties of the Green's function throughout a sector of the complex \beta plane. These estimates are derived in a companion paper [math-ph/0205028].