Electronic Communications in Probability

Quenched central limit theorem in a corner growth setting

H. Christian Gromoll, Mark W. Meckes, and Leonid Petrov

Full-text: Open access


We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 101, 12 pp.

Received: 3 August 2018
Accepted: 29 November 2018
First available in Project Euclid: 19 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

last passage percolation central limit theorem concentration of measure

Creative Commons Attribution 4.0 International License.


Gromoll, H. Christian; Meckes, Mark W.; Petrov, Leonid. Quenched central limit theorem in a corner growth setting. Electron. Commun. Probab. 23 (2018), paper no. 101, 12 pp. doi:10.1214/18-ECP201. https://projecteuclid.org/euclid.ecp/1545188961

Export citation


  • [1] T. Alberts, K. Khanin, and J. Quastel, Intermediate disorder regime for 1+ 1 dimensional directed polymers, Ann. Probab. 42 (2014), no. 3, 1212–1256, arXiv:1202.4398 [math.PR].
  • [2] J. Baik, P. Deift, and T. Suidan, Combinatorics and random matrix theory, Graduate Studies in Mathematics, vol. 172, AMS, 2016.
  • [3] A. Borodin and I. Corwin, Macdonald processes, Probab. Theory Relat. Fields 158 (2014), 225–400, arXiv:1111.4408 [math.PR].
  • [4] A. Borodin, I. Corwin, and P. Ferrari, Free energy fluctuations for directed polymers in random media in 1+ 1 dimension, Comm. Pure Appl. Math. 67 (2014), no. 7, 1129–1214, arXiv:1204.1024 [math.PR].
  • [5] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), arXiv:1106.1596 [math.PR].
  • [6] C.-G. Esseen, On mean central limit theorems, Kungl. Tekn. Högsk. Handl. Stockholm 121 (1958), 31 pp.
  • [7] L. Goldstein, $l^1$ bounds in normal approximation, Ann. Probab. 35 (2007), no. 5, 1888–1930, arXiv:0710.3262 [math.PR].
  • [8] A. Guionnet and O. Zeitouni, Concentration of the spectral measure for large matrices, Electron. Commun. Probab. 5 (2000), 119–136.
  • [9] D. Huse and C. Henley, Pinning and roughening of domain wall in ising systems due to random impurities, Phys. Rev. Lett 54 (1985), no. 25, 2708.
  • [10] J. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment, J. Statist. Phys. 52 (1988), no. 3-4, 609–626.
  • [11] K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. 209 (2000), no. 2, 437–476, arXiv:math/9903134 [math.CO].
  • [12] M. Ledoux, On talagrand’s deviation inequalities for product measures, ESAIM: Probability and statistics 1 (1997), 63–87.
  • [13] M. Meckes, Some results on random circulant matrices, High dimensional probability V: the Luminy volume, IMS, 2009, arXiv:0902.2472 [math.PR], pp. 213–223.
  • [14] N. O’Connell and J. Ortmann, Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights., Electron. J. Probab. 20 (2015), no. 25, 1–18, arXiv:1408.5326 [math.PR].
  • [15] T. Seppäläinen, Lecture notes on the corner growth model, Unpublished lecture notes (2009).
  • [16] T. Seppäläinen, Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40(1) (2012), 19–73, arXiv:0911.2446 [math.PR].
  • [17] T. Seppäläinen, Variational formulas, busemann functions, and fluctuation exponents for the corner growth model with exponential weights, arXiv preprint (2017), arXiv:1709.05771 [math.PR].
  • [18] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Publications Mathematiques de l’IHES 81 (1995), no. 1, 73–205, arXiv:math/9406212 [math.PR].
  • [19] C. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159 (1994), no. 1, 151–174, arXiv:hep-th/9211141.