The mixing time of the lozenge tiling Glauber dynamics
Annales Henri Lebesgue, Volume 6 (2023), pp. 907-940.


Keywords Mixing time, lozenge tilings, random interfaces, dimer model


The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time T mix . In the (d+1)-dimensional setting, d2, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [Hen97, Spo93] and simulations (see [Des02] and the references in [Hen97, Wil04]), one expects convergence to equilibrium to occur on time-scales of order δ -2 in any dimension, with δ0 the lattice mesh.

We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as (2+1)-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [CKP01], the height function concentrates as δ0 around a deterministic profile ϕ, the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture T mix =δ -2+o(1) had been proven, so far, only in the situation where ϕ is an affine function [CMT12]. In this work, we prove the conjecture under the sole assumption that the limit shape ϕ contains no frozen regions (facets).


[ADPZ20] Astala, Kari; Duse, Erik; Prause, István; Zhong, Xiao Dimer models and conformal structures (2020) (

[Agg19] Aggarwal, Amol Universality of tiling local statistics (2019) (, to appear in Annals of Mathematics)

[CKP01] Cohn, Henry; Kenyon, Richard; Propp, James A variational principle for domino tilings, J. Am. Math. Soc., Volume 14 (2001), pp. 297-346 | DOI | MR | Zbl

[CLL22] Caputo, Pietro; Labbé, Cyril; Lacoin, Hubert Spectral gap and cutoff phenomenon for the Gibbs sampler of φ interfaces with convex potential, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 58 (2022) no. 2, pp. 794-826 | MR | Zbl

[CLP98] Cohn, Henry; Larsen, Michael; Propp, James The Shape of a Typical Boxed Plane Partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR | Zbl

[CMT11] Caputo, Pietro; Martinelli, Fabio; Toninelli, Fabio L. Convergence to equilibrium of biased plane partitions, Random Struct. Algorithms, Volume 39 (2011), pp. 83-114 | DOI | MR | Zbl

[CMT12] Caputo, Pietro; Martinelli, Fabio; Toninelli, Fabio L. Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach, Commun. Math. Phys., Volume 311 (2012), pp. 157-189 | DOI | MR | Zbl

[Des02] Destainville, Nicolas Flip dynamics in octagonal rhombus tiling sets, Phys. Rev. Lett., Volume 88 (2002) no. 3, 030601 | DOI

[GG23] Ganguly, Shirshendu; Gheissari, Reza Cutoff for the Glauber dynamics of the lattice free field, Probab. Math. Phys., Volume 4 (2023), pp. 433-475 | DOI | MR | Zbl

[Gia83] Giaquinta, Mariano Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, 105, Princeton University Press, 1983 | Zbl

[GPR09] Greenberg, Sam; Pascoe, Amanda; Randall, Dana Sampling biased lattice configurations using exponential metrics, Proc. of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press (2009), pp. 76-85 | Zbl

[Hen97] Henley, Christopher L. Relaxation time for a dimer covering with height representation, J. Stat. Phys., Volume 89 (1997) no. 3-4, pp. 483-507 | DOI | MR | Zbl

[Ken09] Kenyon, Richard Lectures on dimers (2009) (

[KL98] Kipnis, Claude; Landim, Claudio Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften, 320, Springer, 1998 | Zbl

[KO07] Kenyon, Richard; Okounkov, Andrei Limit shapes and the complex Burgers equation, Acta Math., Volume 199 (2007) no. 2, pp. 263-302 | DOI | MR | Zbl

[Kry08] Krylov, Nicolaĭ V. Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, 96, American Mathematical Society, 2008 | DOI | Zbl

[Lac16] Lacoin, Hubert Mixing time and cutoff for the adjaent transposition shuffle and the simple exclusion, Ann. Probab., Volume 44 (2016) no. 2, pp. 1426-1487 | Zbl

[Las21] Laslier, Benoît Central limit theorem for lozenge tilings with curved limit shape (2021) (

[LL19] Labbé, Cyril; Lacoin, Hubert Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling, Ann. Probab., Volume 47 (2019) no. 3, pp. 1541-1586 | MR | Zbl

[LP17] Levin, David A.; Peres, Yuval Markov chains and mixing times, American Mathematical Society, 2017 | DOI | Zbl

[LRS01] Luby, Michael; Randall, Dana; Sinclair, Alistair Markov Chain Algorithms for Planar Lattice Structures, SIAM J. Comput., Volume 31 (2001), pp. 167-192 | DOI | MR | Zbl

[LT15a] Laslier, Benoît; Toninelli, Fabio L. How quickly can we sample a uniform domino tiling of the 2L×2L square?, Probab. Theory Relat. Fields, Volume 161 (2015) no. 3-4, pp. 509-559 | DOI | Zbl

[LT15b] Laslier, Benoît; Toninelli, Fabio L. Lozenge tilings, Glauber dynamics and macroscopic shape, Commun. Math. Phys., Volume 338 (2015) no. 3, pp. 1287-1326 | DOI | MR | Zbl

[Spo93] Spohn, Herbert Interface motion in models with stochastic dynamics, J. Stat. Phys., Volume 71 (1993), pp. 1081-1132 | DOI | MR | Zbl

[Wil04] Wilson, David B. Mixing times of Lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab., Volume 14 (2004), pp. 274-325 | MR | Zbl