Publication : t15/062

Dimers on Rail Yard Graphs

Boutillier C. (Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris)
Bouttier J. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Chapuy G. (LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13)
Corteel S. (LIAFA, CNRS et Université Paris Diderot, Case 7014, F-75205 Paris Cedex 13)
Ramassamy S. (Mathematics Department, Brown University)
Abstract:
We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson-fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions.
Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick's theorem.
Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimer model. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.
Année de publication : 2017
Revue : Ann. Inst. Henri Poincaré Comb. Phys. Interact 4 479-539 (2017)
DOI : 10.4171/AIHPD/46
Preprint : arXiv:1504.05176
Langue : Anglais

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