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.

39 pages

Cited by in corpus (11)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• Critical exponents for long-range O(n) models below the upper critical dimension
• Functional integral representations for self-avoiding walk
• A renormalisation group method. I. Gaussian integration and normed algebras
• A renormalisation group method. V. A single renormalisation group step
• Joint density for the local times of continuous-time Markov chains
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
• Number variance for hierarchical random walks and related fluctuations
• Mean-field tricritical polymers