The Annals of Applied Probability

Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder

Arjun Krishnan and Jeremy Quastel

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

The free-energy fluctuations of the discrete directed polymer in $1+1$ dimensions is conjecturally in the Tracy–Widom universality class at all finite temperatures and in the intermediate disorder regime. Seppäläinen’s log-gamma polymer was proven to have GUE Tracy–Widom fluctuations in a restricted temperature range by Borodin, Corwin and Remenik [Comm. Math. Phys. 324 (2013) 215–232]. We remove this restriction, and extend this result into the intermediate disorder regime. This result also identifies the scale of fluctuations of the log-gamma polymer in the intermediate disorder regime, and thus verifies a conjecture of Alberts, Khanin and Quastel [Ann. Probab. 42 (2014) 1212–1256]. Using a perturbation argument, we show that any polymer that matches a certain number of moments with the log-gamma polymer also has Tracy–Widom fluctuations in intermediate disorder.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 6 (2018), 3736-3764.

Dates
Received: October 2016
Revised: April 2018
First available in Project Euclid: 8 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1538985634

Digital Object Identifier
doi:10.1214/18-AAP1404

Mathematical Reviews number (MathSciNet)
MR3861825

Zentralblatt MATH identifier
06994405

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments

Keywords
Tracy–Widom distribution universality log-gamma directed polymer

Citation

Krishnan, Arjun; Quastel, Jeremy. Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder. Ann. Appl. Probab. 28 (2018), no. 6, 3736--3764. doi:10.1214/18-AAP1404. https://projecteuclid.org/euclid.aoap/1538985634


Export citation

References

  • [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, DC.
    Mathematical Reviews (MathSciNet): MR0167642
    Zentralblatt MATH: 0171.38503
  • [2] Alberts, T., Khanin, K. and Quastel, J. (2010). Intermediate disorder regime for directed polymers in dimension $1+1$. Phys. Rev. Lett. 105 090603.
  • [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.
    Zentralblatt MATH: 1292.82014
    Digital Object Identifier: doi:10.1214/13-AOP858
    Project Euclid: euclid.aop/1395838128
  • [4] Auffinger, A. (2015). Universality of the polymer model in intermediate disorder. Personal communication.
  • [5] Auffinger, A., Baik, J. and Corwin, I. (2012). Universality for directed polymers in thin rectangles. Available at arXiv:1204.4445 [math-ph].
    arXiv: 1204.4445
  • [6] Auffinger, A. and Chen, W.-K. (2016). Universality of chaos and ultrametricity in mixed $p$-spin models. Comm. Pure Appl. Math. 69 2107–2130. Available at arXiv:1410.8123 [math].
    arXiv: 1410.8123
  • [7] Auffinger, A. and Damron, M. (2013). The scaling relation $\chi=2\xi-1$ for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 10 857–880.
    Zentralblatt MATH: 1277.82029
  • [8] 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.
    Mathematical Reviews (MathSciNet): MR1682248
    Zentralblatt MATH: 0932.05001
    Digital Object Identifier: doi:10.1090/S0894-0347-99-00307-0
  • [9] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
    Zentralblatt MATH: 0944.60003
  • [10] 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.
  • [11] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
  • [12] Borodin, A., Corwin, I., Ferrari, P. and Vető, B. (2015). Height fluctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18 Art. 20.
    Zentralblatt MATH: 1332.82068
    Digital Object Identifier: doi:10.1007/s11040-015-9189-2
  • [13] Borodin, A., Corwin, I. and Remenik, D. (2013). Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Comm. Math. Phys. 324 215–232.
    Zentralblatt MATH: 06222555
    Digital Object Identifier: doi:10.1007/s00220-013-1750-x
  • [14] Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431–457.
    Zentralblatt MATH: 1015.60100
    Digital Object Identifier: doi:10.1007/s004400200213
  • [15] Chatterjee, S. (2005). A simple invariance theorem. Available at arXiv:math/0508213.
  • [16] Chatterjee, S. (2013). The universal relation between scaling exponents in first-passage percolation. Ann. of Math. (2) 177 663–697.
    Zentralblatt MATH: 1271.60101
    Digital Object Identifier: doi:10.4007/annals.2013.177.2.7
  • [17] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
    Zentralblatt MATH: 1042.60069
    Digital Object Identifier: doi:10.3150/bj/1066223275
    Project Euclid: euclid.bj/1066223275
  • [18] Comets, F. and Yoshida, N. (2006). Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 1746–1770.
    Zentralblatt MATH: 1104.60061
    Digital Object Identifier: doi:10.1214/009117905000000828
    Project Euclid: euclid.aop/1163517222
  • [19] Genovese, G. (2012). Universality in bipartite mean field spin glasses. J. Math. Phys. 53 123304.
    Zentralblatt MATH: 1278.82063
    Digital Object Identifier: doi:10.1063/1.4768708
  • [20] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
    Zentralblatt MATH: 1004.82004
    Digital Object Identifier: doi:10.1007/s00220-002-0699-y
  • [21] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
    Zentralblatt MATH: 0969.15008
    Digital Object Identifier: doi:10.1007/s002200050027
  • [22] Lindeberg, J. W. (1920). Über das Exponentialgesetz in der Wahrscheinlichkeitsrechnung. Suomal. Tiedeakat., Helsinki.
    Zentralblatt MATH: 47.0485.01
  • [23] Lindeberg, J. W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15 211–225.
    Zentralblatt MATH: 48.0602.04
    Digital Object Identifier: doi:10.1007/BF01494395
  • [24] O’Connell, N. (2012). Directed polymers and the quantum Toda lattice. Ann. Probab. 40 437–458.
    Zentralblatt MATH: 1245.82091
    Digital Object Identifier: doi:10.1214/10-AOP632
    Project Euclid: euclid.aop/1332772711
  • [25] Seppäläinen, T. (2012). Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 19–73.
  • [26] Speed, T. P. (1983). Cumulants and partition lattices. Aust. J. Stat. 25 378–388.
    Zentralblatt MATH: 0538.60023
    Digital Object Identifier: doi:10.1111/j.1467-842X.1983.tb00391.x
  • [27] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
    Zentralblatt MATH: 1137.82010
    Digital Object Identifier: doi:10.4007/annals.2006.163.221

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more