Real Space Renormalization-Group for Configurational Random Walk Models on a Hierarchical Lattice. The Asymptotic End-to-End Distance of a Weakly SARW in Dimension Four

TL;DR

This paper develops a real space renormalization-group approach to analyze the asymptotic end-to-end distance of weakly self-avoiding random walks in four dimensions on a hierarchical lattice, providing insights into their large-scale behavior.

Contribution

It introduces a novel renormalization-group map for random walk probabilities on hierarchical lattices, specifically addressing weakly self-avoiding walks in four dimensions.

Findings

Derived the asymptotic behavior of the end-to-end distance.

Established the effectiveness of the renormalization-group method for this model.

Provided theoretical insights into self-avoiding walk properties in critical dimension.

Abstract

We present a real space renormalization-group map for probabilities of random walks on a hierarchical lattice. From this, we study the asymptotic behavior of the end-to-end distance of a weakly self- avoiding random walk (SARW) that penalizes the (self-)intersection of two random walks in dimension four on the hierarchical lattice.