LAUSR

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 → 0 limit of ϕ 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 ϕ 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 e 5 where e = 4 − d . In particular, in d = 3 our theory gives z LERW ( d = 3 ) = 1.6243 ± 0.001 , in excellent agreement with the estimate z = 1.62400 ± 0.00005 of numerical simulations.