no. 6
Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
[Forme limite et fluctuations de hauteur pour les couplages parfaits aléatoires des graphes carrés-hexagones]
Boutillier, Cédric1 ; Li, Zhongyang2
1 Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, CNRS 4 place Jussieu 75005 Paris (France)
2 Department of Mathematics University of Connecticut Storrs, Connecticut 06269-3009 (USA)
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×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×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.

Reçu le :
Accepté le :
Accepté après révision le :
Première publication :
Publié le :
DOI : 10.5802/aif.3442
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

Boutillier, Cédric 1 ; Li, Zhongyang 2

1 Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, CNRS 4 place Jussieu 75005 Paris (France)
2 Department of Mathematics University of Connecticut Storrs, Connecticut 06269-3009 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_6_2305_0,
     author = {Boutillier, C\'edric and Li, Zhongyang},
     title = {Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices},
     journal = {Annales de l'Institut Fourier},
     pages = {2305--2386},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {6},
     year = {2021},
     doi = {10.5802/aif.3442},
     zbl = {07554449},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3442/}
}
TY  - JOUR
AU  - Boutillier, Cédric
AU  - Li, Zhongyang
TI  - Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 2305
EP  - 2386
VL  - 71
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3442/
DO  - 10.5802/aif.3442
LA  - en
ID  - AIF_2021__71_6_2305_0
ER  - 
%0 Journal Article
%A Boutillier, Cédric
%A Li, Zhongyang
%T Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
%J Annales de l'Institut Fourier
%D 2021
%P 2305-2386
%V 71
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3442/
%R 10.5802/aif.3442
%G en
%F AIF_2021__71_6_2305_0
Boutillier, Cédric; Li, Zhongyang. Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices. Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2305-2386. doi : 10.5802/aif.3442. https://aif.centre-mersenne.org/articles/10.5802/aif.3442/

[1] Borodin, Alexei Periodic Schur process and cylindrical partitions, Duke Math. J., Volume 140 (2007) no. 3, pp. 391-468 | Zbl

[2] Borodin, Alexei Schur dynamics of the Schur processes, Adv. Math., Volume 228 (2011) no. 4, pp. 2268-2291 | DOI | MR | Zbl

[3] Borodin, Alexei; Ferrari, Patrik L. Anisotropic growth of random surfaces in 2+1 dimensions, Commun. Math. Phys., Volume 325 (2014) no. 2, pp. 603-684 | DOI | MR | Zbl

[4] Borodin, Alexei; Ferrari, Patrik L. Random tilings and Markov chains for interlacing particles, Markov Process. Relat. Fields, Volume 24 (2018) no. 3, pp. 419-451 | MR | Zbl

[5] Boutillier, Cédric; Bouttier, Jérémie; Chapuy, Guillaume; Corteel, Sylvie; Ramassamy, Sanjay Dimers on rail yard graphs, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD), Volume 4 (2017) no. 4, pp. 479-539 | DOI | MR | Zbl

[6] Bufetov, Alexey; Gorin, Vadim Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., Volume 25 (2015) no. 3, pp. 763-814 | DOI | MR | Zbl

[7] Bufetov, Alexey; Gorin, Vadim Fluctuations of particle systems determined by Schur generating functions, Adv. Math., Volume 338 (2018), pp. 702-781 | DOI | MR | Zbl

[8] Bufetov, Alexey; Knizel, Alisa Asymptotics of random domino tilings of rectangular Aztec diamonds, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 3, pp. 1250-1290 | MR | Zbl

[9] Chhita, Sunil; Johansson, Kurt Domino statistics of the two-periodic Aztec diamond, Adv. Math., Volume 294 (2016), pp. 37-149 | DOI | MR | Zbl

[10] Cohn, Henry; Kenyon, Richard; Propp, James A variational principle for domino tilings, J. Am. Math. Soc., Volume 14 (2000) no. 2, pp. 297-346 | DOI | MR | Zbl

[11] Di Francesco, Philippe; Soto-Garrido, Rodrigo Arctic curves of the octahedron equation, J. Phys. A, Math. Theor., Volume 47 (2014) no. 28, 285204, 34 pages | MR | Zbl

[12] Duits, Maurice Gaussian free field in an interlacing particle system with two jump rates, Commun. Pure Appl. Math., Volume 66 (2013) no. 4, pp. 600-643 | DOI | MR | Zbl

[13] Duits, Maurice On global fluctuations for non-colliding processes, Ann. Probab., Volume 46 (2018) no. 3, pp. 1279-1350 | MR | Zbl

[14] Duse, Erik; Metcalfe, Anthony Asymptotic geometry of discrete interlaced patterns. I., Int. J. Math., Volume 26 (2015) no. 11, 1550093, 66 pages | MR | Zbl

[15] Duse, Erik; Metcalfe, Anthony Universalité au bord pour la fluctuation de systèmes discrets de particules entrelacées, Ann. Math. Blaise Pascal, Volume 25 (2018) no. 1, pp. 75-197 | Zbl

[16] Gorin, Vadim; Panova, Greta Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory, Ann. Probab., Volume 43 (2015) no. 6, pp. 3052-3132 | MR | Zbl

[17] Goulden, Ian P.; Guay-Paquet, Mathieu; Novak, Jonathan Monotone Hurwitz numbers and the HCIZ integral, Ann. Math. Blaise Pascal, Volume 21 (2014) no. 1, pp. 71-89 | DOI | Numdam | MR | Zbl

[18] Guionnet, Alice; Zeitouni, Ofer Large deviations asymptotics for spherical integrals, J. Funct. Anal., Volume 188 (2002) no. 2, pp. 461-515 | DOI | MR | Zbl

[19] Harish-Chandra Differential operators on a semisimple Lie algebra, Am. J. Math., Volume 79 (1957), pp. 87-120 | DOI | MR | Zbl

[20] Itzykson, Claude; Zuber, Jean Bernard The planar approximation. II, J. Math. Phys., Volume 21 (1980) no. 3, pp. 411-421 | DOI | MR

[21] Johansson, Kurt The arctic circle boundary and the Airy process, Ann. Probab., Volume 33 (2005) no. 1, pp. 1-30 | MR | Zbl

[22] Johansson, Kurt; Nordenstam, Eric Eigenvalues of GUE Minors, Electron. J. Probab., Volume 11 (2006), pp. 1342-1371 | MR | Zbl

[23] Jokusch, Wiliam; Propp, James; Shor, Peter Random domino tilings and the arctic circle theorem (1998) (https://arxiv.org/abs/math/9801068)

[24] Kasteleyn, P. W. The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice, Physica, Volume 27 (1961), pp. 1209-1225 | Zbl

[25] Kenyon, Richard Conformal invariance of domino tiling, Ann. Probab., Volume 28 (2000) no. 2, pp. 759-795 | MR | Zbl

[26] Kenyon, Richard Dominos and the Gaussian free field, Ann. Probab., Volume 29 (2001) no. 3, pp. 1128-1137 | MR | Zbl

[27] Kenyon, Richard; Okounkov, Andrei Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | DOI | MR | Zbl

[28] Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott Dimers and Amoebae, Ann. Math., Volume 163 (2006) no. 3, pp. 1019-1056 | DOI | MR | Zbl

[29] Li, Zhongyang Conformal invariance of dimer heights on isoradial double graphs, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD), Volume 4 (2017) no. 3, pp. 273-307 | MR | Zbl

[30] Li, Zhongyang Fluctuations of dimer heights on contracting square-hexagon lattices (2018) (https://arxiv.org/abs/1809.08727)

[31] Li, Zhongyang Schur function at general points and limit shape of perfect matchings on contracting square hexagon lattices with piecewise boundary conditions (2018) (https://arxiv.org/abs/1807.06175)

[32] Li, Zhongyang Asymptotics of Schur functions on almost staircase partitions, Electron. Commun. Probab., Volume 25 (2020), 51, 13 pages | MR | Zbl

[33] Macdonald, Ian G. Symmetric Functions and Hall Polynomials, Oxford Science Publications, Oxford University Press, 1998

[34] Mehta, Madan L. Random Matrices, Pure and Applied Mathematics, 142, Elsevier, 2004 | MR

[35] Mkrtchyan, Sevak; Petrov, Leonid GUE corners limit of q-distributed lozenge tilings, Electron. J. Probab., Volume 22 (2017), 101, 24 pages | MR | Zbl

[36] Novak, Jonathan Lozenge tilings and Hurwitz numbers, J. Stat. Phys., Volume 161 (2015) no. 2, pp. 509-517 | DOI | MR | Zbl

[37] Okounkov, Andrei Toda equations for Hurwitz numbers, Math. Res. Lett., Volume 7 (2000) no. 4, pp. 447-453 | DOI | MR | Zbl

[38] Okounkov, Andrei; Reshetikhin, Nicolai Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Am. Math. Soc., Volume 16 (2003) no. 3, pp. 581-603 | DOI | MR | Zbl

[39] Okounkov, Andrei; Reshetikhin, Nicolai The birth of a random matrix, Mosc. Math. J., Volume 6 (2006) no. 3, pp. 553-566 | DOI | MR | Zbl

[40] Okounkov, Andrei; Reshetikhin, Nicolai Random skew plane partitions and Pearcey process, Commun. Math. Phys., Volume 269 (2007), pp. 571-609 | DOI | MR | Zbl

[41] Percus, Jerome K. One more technique for the dimer problem, J. Math. Phys., Volume 10 (1969), p. 1881 | DOI | MR

[42] Petrov, Leonid Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field, Ann. Probab., Volume 43 (2015) no. 1, pp. 1-43 | MR | Zbl

[43] Sheffield, Scott Gaussian free fields for mathematicians, Probab. Theory Relat. Fields, Volume 139 (2007) no. 3-4, pp. 521-541 | DOI | MR | Zbl

[44] Thurston, William P. Conway’s tiling groups, Am. Math. Mon., Volume 97 (1990) no. 8, pp. 757-773 | DOI | MR | Zbl

Cité par Sources :