Electronic Journal of Probability

Subgaussian concentration and rates of convergence in directed polymers

Kenneth Alexander and Nikolaos Zygouras

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Abstract

We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder.  We study the partition function $Z_{N,\omega}$ and establish exponential concentration of $\log Z_{N,\omega}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O( \sqrt{\frac{N}{\log N}}\log \log N)$

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 5, 28 pp.

Dates
Accepted: 11 January 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064230

Digital Object Identifier
doi:10.1214/EJP.v18-2005

Mathematical Reviews number (MathSciNet)
MR3024099

Zentralblatt MATH identifier
1281.82036

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
directed polymers concentration modified Poincar\'e inequalities coarse graining

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Alexander, Kenneth; Zygouras, Nikolaos. Subgaussian concentration and rates of convergence in directed polymers. Electron. J. Probab. 18 (2013), paper no. 5, 28 pp. doi:10.1214/EJP.v18-2005. https://projecteuclid.org/euclid.ejp/1465064230


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References

  • Alexander, Kenneth S. Lower bounds on the connectivity function in all directions for Bernoulli percolation in two and three dimensions. Ann. Probab. 18 (1990), no. 4, 1547–1562.
  • Alexander, Kenneth S. A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993), no. 1, 81–90.
  • Alexander, Kenneth S. Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 (1997), no. 1, 30–55.
  • Alexander, Kenneth S. Power-law corrections to exponential decay of connectivities and correlations in lattice models. Ann. Probab. 29 (2001), no. 1, 92–122.
  • Alexander, K. S. Subgaussian rates of convergence of means in directed first passage percolation. (2011) arXiv:1101.1549
  • Borodin, A., Corwin, I., Remenik, D. Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity (2012) arXiv:1206.4573
  • Benaïm, Michel; Rossignol, Raphael. Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 3, 544–573.
  • Benjamini, Itai; Kalai, Gil; Schramm, Oded. First passage percolation has sublinear distance variance. Ann. Probab. 31 (2003), no. 4, 1970–1978.
  • Chatterjee, S., The universal relation between scaling exponents in first-passage percolation, arXiv:1105.4566
  • Chatterjee, S., Disorder chaos and multiple valleys in spin glasses, (2009) arXiv:0907.3381
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705–723.
  • Comets, Francis; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006), no. 5, 1746–1770.
  • Corwin, I., O'Connel, N.M., Seppäläinen, T., Zygouras, N., Tropical combinatorics and Whitaker functions, (2011) arXiv:1110.3489
  • Graham, B. Sublinear variance for directed last passage percolation. Journal Th. Prob. (2010)
  • Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
  • Kardar, M., Parisi, G., Zhang, Y.C., Dynamic scaling of growing interfaces phPhys. Rev. Lett. 56 (1986), 889–892
  • Kesten, Harry. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993), no. 2, 296–338.
  • Krug, J., Spohn, H., Kinetic roughening of growing surfaces ph In Solids Far From Equilibrium: Growth, Morphology and Defects (C. Godreche, ed.). Cambridge Univ. Press. (1991)
  • Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9
  • Wüthrich, Mario V. Scaling identity for crossing Brownian motion in a Poissonian potential. Probab. Theory Related Fields 112 (1998), no. 3, 299–319.
  • Seppäläinen, Timo. Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40 (2012), no. 1, 19–73.
  • Piza, M. S. T. Directed polymers in a random environment: some results on fluctuations. J. Statist. Phys. 89 (1997), no. 3-4, 581–603.