Field Theories for Loop-Erased Random Walks

Kay Joerg Wiese, Andrei A. Fedorenko

Published 2018-02-24Version 1

We analyze candidate field theories for loop-erased random walks (LERWs) in dimensions $2\le d\le 4$. The first such candidate is $\phi^4$-theory with $O(n)$-symmetry at $n=-2$. The link is established via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder. Here the depinning transition is described by a non-analytic fixed point whose relation to the LERW had been conjectured earlier using analogies with Abelian sandpiles. We show diagrammatically order by order in the coupling constant that both theories yield identical results for key quantities such as the renormalization-group $\beta$-function, and the scaling dimensions of the observables which we identify with the fractal dimension of LERWs. While in $\phi^4$ theory the latter is obtained from the crossover exponent encoded in the operator $\phi_1\phi_2$, in the charge-density-wave formulation it is given by the dynamical exponent $z$. The formal equivalence between the two theories is explicitly checked to 4-loop order. For the fractal dimension of LERWs in $d=3$ it gives at 5-loop order $z=1.624\pm 0.002$, in agreement with the prediction $z = 1.624 00 \pm 0.00005$ of numerical simulations. We also show that a minimal description of LERWs can be formulated in terms of complex fermions. These three models constitute a hierarchy of field theories for LERWs.