Field theories for loop-erased random walks

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 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 zLERW(d = 3) = 1.6243 +/- 0.001, in excellent agreement with the estimate z = 1.62400 +/- 0.00005 of numerical simulations. (C) 2019 The Authors. Published by Elsevier B.V.