Electronic Journal of Probability

Dynamics of lattice triangulations on thin rectangles

Pietro Caputo, Fabio Martinelli, Alistair Sinclair, and Alexandre Stauffer

Full-text: Open access

Abstract

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda ^{|\sigma |}$ where $\lambda >0$ is a parameter and $|\sigma |$ denotes the total edge length of the triangulation. When $\lambda \in (0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp (\Omega (n^2))$ for $\lambda >1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $\lambda =1$.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 29, 22 pp.

Dates
Received: 22 May 2015
Accepted: 4 April 2016
First available in Project Euclid: 14 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1460652930

Digital Object Identifier
doi:10.1214/16-EJP4321

Mathematical Reviews number (MathSciNet)
MR3492933

Zentralblatt MATH identifier
1336.60183

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
lattice triangulation Glauber dynamics mixing times

Rights
Creative Commons Attribution 4.0 International License.

Citation

Caputo, Pietro; Martinelli, Fabio; Sinclair, Alistair; Stauffer, Alexandre. Dynamics of lattice triangulations on thin rectangles. Electron. J. Probab. 21 (2016), paper no. 29, 22 pp. doi:10.1214/16-EJP4321. https://projecteuclid.org/euclid.ejp/1460652930


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References

  • [1] Emile E. Anclin. An upper bound for the number of planar lattice triangulations. Journal of Combinatorial Theory, Series A, 103(2):383–386, August 2003.
  • [2] Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, and Fabio Lucio Toninelli. Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion. Annals of Probability, 42(4):1516–1589, 2014.
  • [3] Pietro Caputo, Fabio Martinelli, Alistair Sinclair, and Alexandre Stauffer. Random lattice triangulations: Structure and algorithms. Annals of Applied Probability, 25(3):1650–1685, 2015. Preliminary version appeared in Proceedings of the 2013 ACM Symposium on Theory of Computing (STOC).
  • [4] Pietro Caputo, Fabio Martinelli, and Fabio Lucio Toninelli. On the approach to equilibrium for a polymer with adsorption and repulsion. Electronic Journal of Probability, 13(10):213–258, 2008.
  • [5] Filippo Cesi. Quasi-factorization of the entropy and logarithmic sobolev inequalities for gibbs random fields. Probability Theory and Related Fields, 120:569–584, 2001.
  • [6] Jesús A. De Loera, Jörg Rambau, and Francisco Santos. Triangulations, volume 25 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2010.
  • [7] Sam Greenberg, Amanda Pascoe, and Dana Randall. Sampling biased lattice configurations using exponential metrics. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 76–85. SIAM, Philadelphia, PA, 2009.
  • [8] Volker Kaibel and Günter M. Ziegler. Counting lattice triangulations. In Surveys in Combinatorics, volume 307 of London Mathematical Society Lecture Note Series, pages 277–307. Cambridge Univ. Press, Cambridge, 2003.
  • [9] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2009.
  • [10] Fabio Martinelli. Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics, pages 93–191. Springer-Verlag, Berlin, Heidelberg, 2004.
  • [11] Fabio Martinelli. Relaxation times of markov chains in statistical mechanics and combinatorial structures. In H. Kesten, editor, Probability on Discrete Structures. Springer-Verlag, Heidelberg, 2004.
  • [12] Alistair Sinclair. Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability and Computing, 1(4):351–370, 1992.
  • [13] Alexandre Stauffer. A Lyapunov function for Glauber dynamics on lattice triangulations, 2015. Preprint at arXiv:1504.07980 [math.PR].