Relation between directed polymers in random media and random bond dimer models

TL;DR

This paper explores the connection between lattice dimer models and continuum elastic models of fluctuating polymers, clarifying their relationship in both deterministic and random media, and resolving discrepancies in previous results.

Contribution

It establishes a detailed equivalence between dimer models and polymer descriptions, including in the presence of randomness, and clarifies the ensemble correspondence.

Findings

Derived polymer density and line tension from dimer weights.

Proved equivalence of ensembles for the hexagonal lattice.

Resolved discrepancies between numerical and theoretical results for random dimers.

Abstract

We reassess the relation between classical lattice dimer models and the continuum elastic description of a lattice of fluctuating polymers. In the absence of randomness we determine the density and line tension of the polymers in terms of the bond weights of hard-core dimers on the square and the hexagonal lattice. For the latter, we demonstrate the equivalence of the canonical ensemble for the dimer model and the grand-canonical description for polymers by performing explicitly the continuum limit. Using this equivalence for the random bond dimer model on a square lattice, we resolve a previously observed discrepancy between numerical results for the random dimer model and a replica approach for polymers in random media. Further potential applications of the equivalence are briefly discussed.