SciPost Phys. 9, 063 (2020) An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson Assaf Shapira1 and Kay Jörg Wiese2 1 Dipartimento di Matematica e Fisica, Università Roma Tre, Largo S.L. Murialdo, 00146, Roma, Italy. 2 Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris, France. Abstract We give a simplified proof for the equivalence of loop-erased random walks to a lat- tice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the d-dimensional hypercubic lattice, at large scales this theory reduces to a scalar φ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field φ(x) C (standard formula- 2 tion) or a nilpotent one satisfying φ(x) = 0. We discuss basic2 properties of the latter formulation, which has distinct advantages in the lattice model. Copyright A. Shapira and K. J. Wiese. Received 16-06-2020 This work is licensed under the Creative Commons Accepted 29-10-2020 Check for Attribution 4.0 International License. Published 04-11-2020 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.9.5.063 Contents 1 Introduction2 2 The loop-erased random walk2 3 Viennot’s theorem3 4 A lattice action with two complex fermions and one complex boson6 5 Trading nilpotent bosons for standard bosons9 6 Selfavoiding polymers, and other applications 11 7 Conclusion and further questions 11 References 12 1 SciPost Phys. 9, 063 (2020) 1 Introduction The loop-erased random walk (LERW) introduced by Lawler in [1] and further developed in [2–4] can in two dimensions be described by Schramm-Löwner evolution (SLE) at κ = 2 [5]. It corresponds to a conformal field theory with central charge c = 2. When reformulated in terms of loops, the lattice O(n)-model can be defined for 2 −n 2 [6] (see also [7], page 187). In the limit of n 2 it has the same conformal− field≤ theory≤ (CFT), a relation which also holds off criticality![8 −], suggesting the description by one complex fermion. The latter, however, does not allow one to assess properties of loop-erased random walks after erasure. Recently, it was established [9, 10] that the equivalence to the O(n) model holds in any dimension d 2, by mapping loop-erased random walks onto the n-component φ4-theory for n 2, or equivalently≥ a theory with two complex fermions, and one complex boson. This formulation! − contains information about the traces of LERWs after erasure, and in particular its Hausdorff dimension. A rigorous proof of the equivalence between LERWs and the O( 2) model was given in Ref. [11]. The ensuing field theory uses nilpotent bosonic fields, i.e. fields− which square to zero. Here we give a simplified proof of the equivalence. We also show that the nilpotent bosonic fields can be replaced by standard bosonic ones. The ensuing field theory has two complex fermions, and one (standard) complex boson, and allows to evaluate, without approximations, observables defined on an arbitrary graph. Keeping only the most relevant terms, it reduces to 4 the φ -type theory with two fermions and one boson given in Refs. [9,10]. Interestingly, there this formulation arose naturally in the analysis of charge-density waves subject to quenched disorder. As a result, the theories discussed here are linked to such diverse systems as charge- density waves, Abelian sandpiles [12, 13], uniform spanning trees [14, 15], the Potts model [14], Laplacian walk [2, 16], and dielectric breakdown [17]. 2 The loop-erased random walk Consider a random walk on a directed graph G. The walk jumps from vertex x to y with rate 2 βx y , and dies out with rate λx = mx . The coefficients βx y x,y G are weights on the graph. In f g 2 P particular, when βx y is positive, G contains an edge from x to y. Denote by rx = λx + y βx y the total rate at which the walk exits from vertex x. We define a path to be a sequence of vertices, denoted ! = (!1,..., !n). We refer to i = 1, ..., n as time. The probability P(!) that the random walk selects the path ! and then stops is λ! ! n q ! , (1) P( ) = r ( ) !n β β!1!2 β!2!3 !n 1!n q ! ... − . (2) ( ) = r r r !1 !2 !n 1 − (a) (a) (a) We call q(!) the weight function. Note that this weight factorizes: If ! = (!1 ,..., !n ) and (b) (b) (b) (a) (b) ! = (!1 ,..., !m ), with !n = !1 , then the composition (a) (b) (a) (a) (b) (b) (a) (b) ! := ! ! = (!1 ,..., !n = !1 ,..., !m ) of the paths ! and ! has weight ◦ a b q(!) = q(!( ))q(!( )). 2 SciPost Phys. 9, 063 (2020) One example to have in mind is d G = Z , (3) λx = λ, (4) βx y = βy x = 1x y . (5) ∼ d This choice describes the simple symmetric nearest-neighbor random walk on Z , stopped at 1 a random time with rate λ; its length has an exponential distribution with mean λ− . A second important example is a random walk in a finite box stopped on the boundary, described by the choice d G = l,..., l , (6) ¨{− g d 0 if x l + 1, . , l 1 , λx = (7) otherwise,2 {− − g 1 βx y = βy x = 1x y . (8) ∼ Given a path ! = (!1,..., !n), we define the loop-erasure procedure. The loop erasure is obtained by applying consecutively the one-loop erasure: look for the first time i at which the path repeats a vertex, so !i = !j for some j < i. The one-loop erasure of ! is the path (!1,..., !j, !i+1,..., !n). We then apply the one-loop erasure to this new path, and continue until the path has no repeating vertices. The resulting path is the loop erasure of !, denoted LE(!). It is self-avoiding, and when the initial path ! is a random walk, it is called the loop- erased random walk (LERW). If γ is a LERW ending at x, the probability to generate it is λx X P(γ) = q(!) . (9) rx !:LE(!)=γ Warning: a self-avoiding path is a cominatorial object, i.e., it has no statistics. The loop-erased random walk is one possible distribution on the set of self-avoiding paths. It should not be confused with the self-avoiding walk or self-avoiding polymer commonly studied in the physics literature, which is another distribution on the set of self-avoiding paths. We distinguish the combinatorial object from the probabilistic one by using the term path. 3 Viennot’s theorem The main tool we use is a combinatorial theorem by Viennot [18]. It is part of the general theory of heaps of pieces. In our case it reduces to a relation between loop-erased random walks, and collections of loops. It allows us to calculate the total weight P(γ) of all paths ! whose loop erasure is γ, and represent the latter as a lattice model. Before discussing the general case, consider a graph consisting of two vertices and one edge, and the following path, γ = x y . (10) The probability that such a path is the result of a loop-erased random walk leaving from x and 3 SciPost Phys. 9, 063 (2020) arriving at y is given by a geometric sum, P(γ) = x y + x y + ... λy βx y βx y βy x βx y = + + ... ry rx rx ry rx λy 1 q γ . (11) = ( ) 1 1 ry 1 rx βx y ry βy x − − − We come back to this example later. A loop is a path ! = (!1,..., !n 1, !n = !1) where the first and last points are identical. − We also require all vertices to be distinct (except !1 and !n), so it cannot be decomposed into smaller loops. Loops obtained from each other via cyclic permutations (dropping the repeated vertex !n) are considered identical. By a collection of disjoint loops we mean a set L = C1, C2,... , each of whose elements is a loop, and the intersection of any pair of loops in L fis empty. Weg denote the set of all such collections by L. In order to formulate the theorem, we fix a self-avoiding path γ. We define the set Lγ to consist of the collections of disjoint loops in which no loop intersects γ. Then Viennot’s theorem can be written as ( L being the number of loops) j j X L Y X X L Y A(γ) := q(γ) ( 1)j j q(C) = q(!) ( 1)j j q(C). (12) L Lγ − C L !:LE(!)=γ × L L − C L 2 2 2 2 On the l.h.s. one sums over the ensemble of collections of loops which do not intersect γ, giving L Q each collection a weight ( 1)j j C L q(C). We later calculate this object using field-theory. The r.h.s. contains two factors.− The2 first is the weight to find the LERW path γ, our object of interest. The second is the partition function X L Y Z := ( 1)j j q(C) (13) L L − C L 2 2 of the loop model on the left-hand side. Assuming the walk to stop at x, this relation can be read as q γ P 1 L Q q C λx X λx ( ) L Lγ ( )j j C L ( ) P(γ) = q(!) = 2 − 2 .