Field Theories for Loop-Erased Random Walks

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 $n\to 0$ limit of ${\varphi }^4$-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 ${\varphi }^4$ 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 $z_{\mathrm{LERW}}(d=3)=1.6243\pm 0.001$, in excellent agreement with the estimate $z=1.62400\pm 0.00005$ of numerical simulations.

18 pages, 223 figures. Added in v2: algebraic proof for the equivalence between the two theories. Explicit diagrammatic expression at 5 loop. v3: final version

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Cited by in corpus (16)

• Precision calculation of critical exponents in the $O(N)$ universality classes with the nonperturbative renormalization group
• Fractal dimension of critical curves in the $O(n)$-symmetric ${\varphi }^4$-model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• Gentle introduction to rigorous Renormalization Group: a worked fermionic example
• Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric ${\varphi }^4$ theory with $n\to -2$ and loop-erased random walks
• Physical networks as network-of-networks
• Renormalization group for measurement and entanglement phase transitions
• Field Theory of Disordered Elastic Interfaces at 3-Loop Order: Critical Exponents and Scaling Functions
• An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson
• Conformal invariance constraints in the $O(N)$ models: a first study within the nonperturbative renormalization group
• Loop-erased random walk as a spin system observable
• Roughness and critical force for depinning at 3-loop order
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• One-point function estimates for loop-erased random walk in three dimensions
• Large Orders and Strong-Coupling Limit in Functional Renormalization
• Non-trivial fixed point of a ${\psi }_d^4$ fermionic theory, II. Anomalous exponent and scaling operators