Dimers on Rail Yard Graphs

• Published in: Annales de l’Institut Henri Poincaré D
• Main Authors: Boutillier, Cédric, Bouttier, Jérémie, Chapuy, Guillaume, Corteel, Sylvie, Ramassamy, Sanjay
• Other Authors: Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Direction de Recherche Fondamentale (CEA) (DRF (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, Brown University, Projet Emergences Combinatoire à Paris (Ville de Paris), Monahan Foundation, ANR-08-JCJC-0011,Icomb(2008), ANR-10-BLAN-0123,MAC2,Modèles aléatoires critiques bi-dimensionnels(2010), ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2017
• Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]; Arctic; Pyramid
• Online Access: https://cea.hal.science/cea-01144118; https://cea.hal.science/cea-01144118/document; https://cea.hal.science/cea-01144118/file/dimerstat.pdf; https://doi.org/10.4171/AIHPD/46

International audience 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.