The Annals of Applied Probability

Equality of critical points for polymer depinning transitions with loop exponent one

Kenneth S. Alexander and Nikos Zygouras

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Abstract

We consider a polymer with configuration modelled by the trajectory of a Markov chain, interacting with a potential of form u+Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length n takes the form φ(n)/n for some slowly varying φ; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ, at least at low temperatures.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 1 (2010), 356-366.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1262962326

Digital Object Identifier
doi:10.1214/09-AAP621

Mathematical Reviews number (MathSciNet)
MR2582651

Zentralblatt MATH identifier
1187.82054

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
Pinning polymer disorder random potential quenched critical point

Citation

Alexander, Kenneth S.; Zygouras, Nikos. Equality of critical points for polymer depinning transitions with loop exponent one. Ann. Appl. Probab. 20 (2010), no. 1, 356--366. doi:10.1214/09-AAP621. https://projecteuclid.org/euclid.aoap/1262962326


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References

  • [1] Alexander, K. S. (2008). The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 117–146.
  • [2] Alexander, K. S. and Sidoravicius, V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 636–669.
  • [3] Alexander, K. S. and Zygouras, N. (2009). Quenched and annealed critical points in polymer pinning models. Comm. Math. Phys. 291 659–689.
  • [4] Derrida, B., Giacomin, G., Lacoin, H. and Toninelli, F. L. (2009). Fractional moment bounds and disorder relevance for pinning models. Comm. Math. Phys. 287 867–887.
  • [5] Derrida, B., Giacomin, G., Lacoin, H. and Toninelli, F. L. (2009). Personal communication.
  • [6] Derrida, B., Hakim, V. and Vannimenus, J. (1992). Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66 1189–1213.
  • [7] Forgács, G., Luck, J. M., Nieuwenhuizen, T. M. and Orland, H. (1988). Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Stat. Phys. 51 29–56.
  • [8] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
  • [9] Giacomin, G. (2009). Renewal sequences, disordered potentials, and pinning phenomena. In Spin Glasses: Statics and Dynamics, Summer School, Paris 2007, Progress in Probability. Birkhauser, Boston. To appear.
  • [10] Giacomin, G., Lacoin, H. and Toninelli, F. L. (2009). Marginal relevance of disorder for pinning models. Comm. Pure Appl. Math. To appear.
  • [11] Giacomin, G. and Toninelli, F. L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions. Comm. Math. Phys. 266 1–16.
  • [12] Giacomin, G. and Toninelli, F. L. (2006). The localized phase of disordered copolymers with adsorption. ALEA Lat. Am. J. Probab. Math. Stat. 1 149–180.
  • [13] Giacomin, G. and Toninelli, F. L. (2007). On the irrelevant disorder regime of pinning models. Ann. Probab. 37 1841–1875.
  • [14] Jain, N. C. and Pruitt, W. E. (1972). The range of random walk. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Probability Theory 3 31–50. Univ. California Press, Berkeley, CA.
  • [15] Toninelli, F. L. (2008). A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 389–401.
  • [16] Toninelli, F. (2009). Localization transition in disordered pinning models. Effect of randomness on the critical properties. In Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics 1970 129–176. Springer, Berlin.
  • [17] Toninelli, F. L. (2008). Disordered pinning models and copolymers: Beyond annealed bounds. Ann. Appl. Probab. 18 1569–1587.