Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices

Boutillier, Cédric; Li, Zhongyang

Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2305-2386.

Nous étudions les couplages parfaits de graphes, construits en prenant, pour chaque ligne, une ligne soit du réseau carré, soit du réseau hexagonal. Étant donnés des poids sur les arêtes avec une période $1\times n$, la fonction de partition est une fonction de Schur dépendant des poids. Nous obtenons dans la limite des grands systèmes une loi des grands nombres (forme limite) et un théorème central limite (convergence vers le champ libre) pour la fonction de hauteur associée. La distribution de certains dimères près du point de contact au bord converge vers celle des valeurs propres de l’ensemble unitaire gaussien. De plus, dans la limite d’échelle de systèmes pour lesquels chaque segment du bord croît linéairement avec la taille du graphe, le bord de la zone gelée est une courbe nuage avec des points de contact sur chaque segment du bord inférieur dont le nombre dépend de la période.

We study perfect matchings on the contracting square-hexagon lattice, constructed row by row either from a row of the square grid or of the hexagonal lattice. Given $1\times n$ periodic weights to edges, we consider the probabilities of dimers proportional to the product of edge weights. We show that the partition function equals a Schur function of the edge weights. We then prove the Law of Large Numbers (limit shape) and the Central Limit Theorem (convergence to the Gaussian free field) for the corresponding height functions. We also show that certain types of dimers near the turning corner converge in distribution to the eigenvalues of Gaussian Unitary Ensemble, and that in the scaling limit when each segment of the bottom boundary grows linearly with respect to the dimension of the graph, the frozen boundary is a cloud curve with multiple tangent points (depending on the period) along each horizontal boundary segment.

Classification : 82B20, 05E05, 74A50, 60B20
Keywords: dimer, perfect matching, limit shape, Gaussian free field, Schur function
Mot clés : dimères, couplage parfait, forme limite, champ libre gaussien, fonction de Schur

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