Field theories for loop-erased random walks

Kay Jörg Wiese,Andrei A Fedorenko

doi:10.1016/j.nuclphysb.2019.114696

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 ϕ4-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 ϕ4 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 ε5 where ε=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.

Highlights

Random walks (RWs) which are not allowed to self-intersect play an important role in combinatorics, statistical physics and quantum field theory
We show below that both the β-function and the fractal dimension of loop-erased random walks (LERWs) coincide
For n = −2, the 2-point function between two points x and z in the φ4 theory is the sum over all LERWs from x to z, weighted by the chemical potential m2 conjugate to its construction time t

Summary

Introduction

Random walks (RWs) which are not allowed to self-intersect play an important role in combinatorics, statistical physics and quantum field theory. For n = −2, the 2-point function between two points x and z in the φ4 theory is the sum over all LERWs from x to z, weighted by the chemical potential m2 conjugate to its construction time t It equals the free propagator from x to z, since coloring loops in red or erasing them does not change the propagator. The additional time difference, when appearing together with a response function, acts by inserting an additional point into the latter, as can be seen from the definition (23), and the relation t tR(k, t) = dt R(k, t )R(k, t − t ) Following this strategy, we checked that up to 4-loop order all diagrams appearing after timeintegration are equivalent to those encountered in expectations of O, defined in Eq (8). This is equivalent to the insertion of O defined above in Eqs. (8) or (9)

Fractal dimension of LERWs at 5-loop order

Summary and Perspectives