Field theory conjecture for loop-erased random walks

TL;DR

This paper proposes that the functional renormalization group (FRG) can serve as a field theoretic framework for loop-erased random walks, enabling systematic calculations of their fractal dimensions and corrections.

Contribution

It introduces a novel FRG-based approach to describe LERW, providing analytical predictions consistent with known bounds and numerical data.

Findings

FRG agrees with rigorous bounds up to two loops

Reproduces leading logarithmic corrections at d=4

Predicts universal subleading logarithmic correction in four dimensions

Abstract

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.