Annals of Probability

Directed polymers in heavy-tail random environment

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

Abstract

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha \in (0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, that is, when the inverse temperature temperature $\beta =\beta_{n}$ vanishes as the size of the system $n$ goes to infinity. When $\alpha \in (1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n}\leq h_{n}\leq n$ can be achieved by tuning properly $\beta_{n}$, allowing to interpolate between all superdiffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [Ann. Probab. 44 (2016) 4006–4048]—we actually identify five different regimes. On the other hand, when $\alpha <1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last-Passage Percolation (E-LPP), introduced in a companion paper [Ann. Appl. Probab. 29 (2019) 1878–1903].

Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 4024-4076.

Dates
Received: May 2018
Revised: January 2019
First available in Project Euclid: 2 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1575277347

Digital Object Identifier
doi:10.1214/19-AOP1353

Mathematical Reviews number (MathSciNet)
MR4038048

Zentralblatt MATH identifier
07212177

Subjects
Primary: 60F05: Central limit and other weak theorems 82D60: Polymers
Secondary: 60K37: Processes in random environments 60G70: Extreme value theory; extremal processes

Keywords
Directed polymer heavy-tail distributions weak-coupling limit last-passage percolation superdiffusivity

Citation

Berger, Quentin; Torri, Niccolò. Directed polymers in heavy-tail random environment. Ann. Probab. 47 (2019), no. 6, 4024--4076. doi:10.1214/19-AOP1353. https://projecteuclid.org/euclid.aop/1575277347


Export citation

References

  • [1] Alberts, T., Khanin, K. and Quastel, J. (2010). Intermediate disorder regime for directed polymers in dimension $1+1$. Phys. Rev. Lett. 105 090603.
  • [2] Alberts, T., Khanin, K. and Quastel, J. (2014). The continuum directed random polymer. J. Stat. Phys. 154 305–326.
  • [3] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension $1+1$. Ann. Probab. 42 1212–1256.
  • [4] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 466–537.
  • [5] Auffinger, A. and Louidor, O. (2011). Directed polymers in a random environment with heavy tails. Comm. Pure Appl. Math. 64 183–204.
  • [6] Balázs, M., Quastel, J. and Seppäläinen, T. (2011). Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 683–708.
  • [7] Berger, Q. and Torri, N. (2019). Entropy-controlled last-passage percolation. Ann. Appl. Probab. 29 1878–1903.
  • [8] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [9] Biroli, G., Bouchaud, J.-P. and Potters, M. (2007). Extreme value problems in random matrix theory and other disordered systems. J. Stat. Mech. Theory Exp. 7 P07019, 15.
  • [10] Caravenna, F., Sun, R. and Zygouras, N. (2017). Polynomial chaos and scaling limits of disordered systems. J. Eur. Math. Soc. (JEMS) 19 1–65.
  • [11] Comets, F. (2017). Directed Polymers in Random Environments. Lecture Notes in Math. 2175. Springer, Cham.
  • [12] 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.
  • [13] 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.
  • [14] Gueudre, T., Le Doussal, P., Bouchaud, J.-P. and Rosso, A. (2015). Ground-state statistics of directed polymers with heavy-tailed disorder. Phys. Rev. E (3) 91 062110, 10.
  • [15] Hambly, B. and Martin, J. B. (2007). Heavy tails in last-passage percolation. Probab. Theory Related Fields 137 227–275.
  • [16] Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708–2711.
  • [17] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). Springer, New York.
  • [18] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
  • [19] Moreno Flores, G. R., Seppäläinen, T. and Valkó, B. (2014). Fluctuation exponents for directed polymers in the intermediate disorder regime. Electron. J. Probab. 19 no. 89, 28.
  • [20] Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Probab. 7 745–789.
  • [21] Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 19–73.
  • [22] Seppäläinen, T. and Valkó, B. (2010). Bounds for scaling exponents for a $1+1$ dimensional directed polymer in a Brownian environment. ALEA Lat. Am. J. Probab. Math. Stat. 7 451–476.
  • [23] 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.
  • [24] Torri, N. (2016). Pinning model with heavy tailed disorder. Stochastic Process. Appl. 126 542–571.