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 massless Gaussian Free Field (GFF), 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 fermionic language, the height difference hΞ - hη takes the form of a non-local operator, consisting of a sum of 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.

Height fluctuations in interacting dimers / A. Giuliani, V. Mastropietro, F.L. Toninelli. - In: ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES. - ISSN 0246-0203. - 53:1(2017), pp. 98-168. [10.1214/15-AIHP710]

Height fluctuations in interacting dimers

V. Mastropietro
Secondo
;
2017

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 massless Gaussian Free Field (GFF), 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 fermionic language, the height difference hΞ - hη takes the form of a non-local operator, consisting of a sum of 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.
Constructive renormalization group; Dimer model; Gaussian free field; Universality; Statistics and Probability; Statistics, Probability and Uncertainty
Settore MAT/07 - Fisica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/491448
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