2012

Asymptotic domino statistics in the Aztec diamond

Chhita, Sunil, Johansson, Kurt, Young, Benjamin

Understand

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes.

A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process.
We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond.
This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott.

Built on

Kasteleyn, P. W.P. W. (1961). The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica 27 1209–1225
1961
Earlier work this paper cites.
Temperley, H. N. V.H. N. V. andFisher, Michael E.M. E. (1961). Dimer problem in statistical mechanics—an exact result. Philos. Mag. (8) 6 1061–1063
1961
Earlier work this paper cites.
Kasteleyn, P. W.P. W. (1963). Dimer statistics and phase transitions. J. Math. Phys. 4 287–293
1963
Earlier work this paper cites.
Levitov, L. S.L. S. (1989). The rvb model as the problem of the surface of a quantum crystal. JETP Lett. 50 469–472
1989
Earlier work this paper cites.
Levitov, L. S.L. S. (1990). Equivalence of the dimer resonating-valence-bond problem to the quantum roughening problem. Phys. Rev. Lett. 64 92–94
1990
Earlier work this paper cites.
Thurston, William P.W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97 757–773
Original
1990
Earlier work this paper cites.
Elkies, NoamN., Kuperberg, GregG., Larsen, MichaelM. andPropp, JamesJ. (1992). Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. 1 111–132
1992
Earlier work this paper cites.
Elkies, NoamN., Kuperberg, GregG., Larsen, MichaelM. andPropp, JamesJ. (1992). Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 219–234
1992
Earlier work this paper cites.
Cohn, HenryH., Elkies, NoamN. andPropp, JamesJ. (1996). Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85 117–166
1996
Earlier work this paper cites.
Kenyon, RichardR. (1997). Local statistics of lattice dimers. Ann. Inst. Henri Poincaré Probab. Stat. 33 591–618
1997
Earlier work this paper cites.
Cohn, HenryH., Larsen, MichaelM. andPropp, JamesJ. (1998). The shape of a typical boxed plane partition. New York J. Math. 4 137–165 (electronic)
1998
Earlier work this paper cites.
Jockusch, W.W., Propp, J.J. andShor, P.P. (1998). Random domino tilings and the Arctic circle theorem. Preprint. Available at \arxivurl
1998
Earlier work this paper cites.
Gohberg, IsraelI., Goldberg, SeymourS. andKrupnik, NahumN. (2000). Traces and Determinants of Linear Operators. Operator Theory: Advances and Applications 116. Birkhäuser, Basel
2000
Earlier work this paper cites.
Helfgott, HaroldH. (2000). Edge effects on local statistics in lattice dimers: A study of the Aztec diamond (finite case). Preprint. Available at \arxivurl
2000
Earlier work this paper cites.
Kenyon, RichardR. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759–795
2000
Earlier work this paper cites.

Similar

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2000
Cited alongside, same era.
Cohn, HenryH., Kenyon, RichardR. andPropp, JamesJ. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346 (electronic)
2001
Cited alongside, same era.
Kenyon, RichardR. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128–1137
2001
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Luby, MichaelM., Randall, DanaD. andSinclair, AlistairA. (2001). Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31 167–192
2001
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Borodin, AlexeiA. (2002). Duality of orthogonal polynomials on a finite set. J. Stat. Phys. 109 1109–1120
2002
Cited alongside, same era.
Johansson, KurtK. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 225–280
2002
Cited alongside, same era.
Okounkov, AndreiA. andReshetikhin, NikolaiN. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581–603 (electronic)
2003
Cited alongside, same era.
Johansson, KurtK. (2005). The arctic circle boundary and the Airy process. Ann. Probab. 33 1–30
2005
Cited alongside, same era.
Johansson, KurtK. (2006). Random matrices and determinantal processes. In Mathematical Statistical Physics 1–55. Elsevier, Amsterdam
2006
Cited alongside, same era.
Johansson, KurtK. andNordenstam, EricE. (2006). Eigenvalues of GUE minors. Electron. J. Probab. 11 1342–1371
2006
Cited alongside, same era.
Kenyon, RichardR., Okounkov, AndreiA. andSheffield, ScottS. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056
2006
Cited alongside, same era.
Boutillier, CédricC. (2007). Pattern densities in non-frozen planar dimer models. Comm. Math. Phys. 271 55–91
2007
Cited alongside, same era.

Then

Daley, D. J.D. J. andVere-Jones, D.D. (2008). An Introduction to the Theory of Point Processes: General Theory and Structure, 2nd ed. Probability and Its Applications (New York) 2. Springer, New York
2008
Later among the works it cites.
Kenyon, RichardR. (2008). Height fluctuations in the honeycomb dimer model. Comm. Math. Phys. 281 675–709
2008
Later among the works it cites.
Kenyon, RichardR. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser. 16 191–230. Amer. Math. Soc., Providence, RI
2009
Later among the works it cites.
Anderson, Greg W.G. W., Guionnet, AliceA. andZeitouni, OferO. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge
2010
Later among the works it cites.
Borodin, AlexeiA., Gorin, VadimV. andRains, Eric M.E. M. (2010). q q -distributions on boxed plane partitions. Selecta Math. (N.S.) 16 731–789
2010
Later among the works it cites.
Borodin, AlexeiA. andShlosman, SenyaS. (2010). Gibbs ensembles of nonintersecting paths. Comm. Math. Phys. 293 145–170
2010
Later among the works it cites.
Dubédat, JulienJ. (2011). Dimers and analytic torsion 1. Available at \arxivurl
Original
2011
Later among the works it cites.
Fleming, Benjamin J.B. J. andForrester, Peter J.P. J. (2011). Interlaced particle systems and tilings of the Aztec diamond. J. Stat. Phys. 142 441–459
2011
Later among the works it cites.
Broomhead, NathanN. (2012). Dimer models and Calabi–Yau algebras. Mem. Amer. Math. Soc. 215 viii+86
2012
Closest in time.
Petrov, LeonidL. (2012). Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes. Available at \arxivurl
Original
2012
Closest in time.
Petrov, LeonidL. (2012). Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Available at \arxivurl
Original
2012
Closest in time.
Romik, DanD. (2012). Arctic circles, domino tilings and square Young tableaux. Ann. Probab. 40 611–647
2012
Closest in time.
Duits, MauriceM. (2013). Gaussian free field in an interlacing particle system with two jump rates. Comm. Pure Appl. Math. 66 600–643
2013
Closest in time.
Borodin, AlexeiA. andFerrari, Patrik L.P. L. (2014). Anisotropic growth of random surfaces in 2 + 1 2+1 dimensions. Comm. Math. Phys. 325 603–684
2014
Closest in time.

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