Authors: A. Giuliani, V. Mastropietro, F. L. Toninelli
Height fluctuations in interacting dimers
Abstract: We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of Z2, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces ξ of Z2. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference hξ-hη converge to those of the Gaussian Free Field, asymptotically as |ξ-η|→∞. We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the lattice language, the height difference hξ-hη takes the form of a non-local fermionic operator, consisting of a sum of fermionic monomials along an arbitrary path connecting ξ and η. As in the non-interacting case, this path-independence plays a crucial role in the proof.
Keywords: Dimers, Kasteleyn's solution, lattice fermion representation, multiscale analysis, critical exponents, Renormalization Group, Gaussain scaling limit.
Dipartimento di Matematica e Fisica, Università di Roma Tre
E-mail address: giuliani AT mat DOT uniroma3 DOT it
Dipartimento di Matematica, Università di Milano
E-mail address: mastropietro AT unimi DOT it
F. L. Toninelli
Institute Cammile Jordan, Université de Lyon
E-mail address: toninelli AT math DOT univ-lyon1 DOT fr