The Annals of Applied Probability

Entropy-controlled Last-Passage Percolation

Quentin Berger and Niccolò Torri

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We introduce a natural generalization of Hammersley’s Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E-LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).

Article information

Ann. Appl. Probab., Volume 29, Number 3 (2019), 1878-1903.

Received: May 2018
Revised: October 2018
First available in Project Euclid: 19 February 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems

Last-passage percolation heavy-tail distributions path entropy


Berger, Quentin; Torri, Niccolò. Entropy-controlled Last-Passage Percolation. Ann. Appl. Probab. 29 (2019), no. 3, 1878--1903. doi:10.1214/18-AAP1448.

Export citation


  • [1] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab. 42 1212–1256.
  • [2] Auffinger, A. and Louidor, O. (2011). Directed polymers in a random environment with heavy tails. Comm. Pure Appl. Math. 64 183–204.
  • [3] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
  • [4] Berger, Q. and Torri, N. (2018). Directed polymers in heavy-tail random environment. Available at arXiv:1802.03355.
  • [5] Berger, Q. and Torri, N. (2018). Beyond Hammersley’s Last-Passage Percolation: A discussion on possible new local and global constraints. Available at ArXiv:1802.04046.
  • [6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [7] Comets, F. (2016). Directed Polymers in Random Environments. Ecole d’Eté de probabilités de Saint-Flour 2175. Springer, Cham.
  • [8] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115–142. Math. Soc. Japan, Tokyo.
  • [9] den Hollander, F. (2007). Random Polymers. Ecole d’Eté de probabilités de Saint-Flour 1974. Springer, Berlin.
  • [10] Dey, P. S. and Zygouras, N. (2016). High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder. Ann. Probab. 44 4006–4048.
  • [11] Hambly, B. and Martin, J. B. (2007). Heavy tails in last-passage percolation. Probab. Theory Related Fields 137 227–275.
  • [12] Hammersley, J. M. (1972). A few seedlings of research. In Proc. Sixth Berkeley Symp. Math. Statist. and Probab. 345–394. Univ. California Press, Berkeley, CA.
  • [13] Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222.
  • [14] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Series in Operations Research and Financial Engineering. Springer, New York.
  • [15] Stone, C. (1967). On local and ratio limit theorems. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2 217–224. Univ. California Press, Berkeley, CA.
  • [16] Ulam, S. M. (1961). Monte Carlo calculations in problems of mathematical physics. In Modern Mathematics for the Engineer: Second Series 261–281. McGraw-Hill, New York.
  • [17] Vershik, A. M. and Kerov, S. V. (1977). Asymptotics of the plancherel measure of the symmetric group and the limiting form of Young tables. Sov. Math., Dokl. 18 527–531.