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

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) 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 √ 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 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 [BI2]. ∗Research supported by NSF Grant DMS-9706166

Cite this paper

@inproceedings{Brydges2008EndtoendDF, title={End-to-end Distance from the Green’s Function for a Hierarchical Self-Avoiding Walk in Four Dimensions}, author={David Brydges and John Z Imbrie}, year={2008} }