Field Theories for Loop-Erased Random Walks

TL;DR

This paper establishes a field-theoretic description of loop-erased random walks (LERWs) in arbitrary dimensions, connecting two candidate theories and providing precise estimates of their fractal dimension consistent with numerical simulations.

Contribution

It demonstrates the equivalence of two different field theories for LERWs and computes the fractal dimension to high order in epsilon expansion.

Findings

Both theories yield identical results up to 4-loop order.

The fractal dimension of LERWs in 3D is estimated as 1.6243 ± 0.001.

Theoretical predictions agree with numerical simulations.

Abstract

Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the $n \to 0$ limit of $\phi^4$-theory with $O(n)$-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the $O(n)$-symmetric $\phi^4$ theory at $n=-2$ whose link to LERWs was known in two dimensions due to conformal field theory. Here it is established in arbitrary dimension via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. We explicitly show that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. This allows us to compute the fractal dimension of LERWs to order $\epsilon^5$ where $\epsilon=4-d$. In particular, in $d=3$ our theory gives $z_{\rm LERW}(d=3)= 1.6243 \pm 0.001$, in excellent agreement with the estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.