The Annals of Applied Probability

Pinning of polymers and interfaces by random potentials

Kenneth S. Alexander and Vladas Sidoravicius

Full-text: Open access


We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by, for example, having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction of its monomers at 0 with high probability. This critical point may differ for the quenched, annealed and deterministic cases. We show that self-averaging occurs, meaning that the quenched free energy and critical point are nonrandom, off a null set. We evaluate the critical point for a deterministic interaction (u without added randomness) and establish our main result that the critical point in the quenched case is strictly smaller. We show that, for every fixed u∈ℝ, pinning occurs at sufficiently low temperatures. If the excursion length distribution has polynomial tails and the interaction does not have a finite exponential moment, then pinning occurs for all u∈ℝ at arbitrary temperature. Our results apply to other mathematically similar situations as well, such as a directed polymer that interacts with a random potential located in a one-dimensional defect, or an interface in two dimensions interacting with a random potential along a wall.

Article information

Ann. Appl. Probab., Volume 16, Number 2 (2006), 636-669.

First available in Project Euclid: 29 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Pinning polymer disorder interface random potential


Alexander, Kenneth S.; Sidoravicius, Vladas. Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 (2006), no. 2, 636--669. doi:10.1214/105051606000000015.

Export citation


  • Abraham, D. B. (1980). Solvable model with a roughening transition for a planar Ising ferromagnet. Phys. Rev. Lett. 44 1165–1168.
  • Azuma, K. (1967). Weighted sums of certain dependent random variables. Tohoku Math. J. 19 357–367.
  • Biskup, M. and den Hollander, F. (1999). A heteropolymer near a linear interface. Ann. Appl. Probab. 9 668–687.
  • Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 1–125.
  • Derrida, B., Hakim, V. and Vannimenus, J. (1992). Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 1189–1213.
  • Dunlop, F. and Topolski, K. (2000). Cassie's law and concavity of wall tension with respect to disorder. J. Statist. Phys. 98 1115–1134.
  • Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
  • Forgacs, G., Luck, J. M., Nieuwenhuizen, T. M. and Orland, H. (1986). Wetting of a disordered substrate: Exact critical behavior in two dimensions. Phys. Rev. Lett. 57 2184–2187.
  • Forgacs, G., Luck, J. M., Nieuwenhuizen, T. M. and Orland, H. (1988). Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Statist. Phys. 51 29–56.
  • Galluccio, S. and Graber, R. (1996). Depinning transition of a directed polymer by a periodic potential: A $d$-dimensional solution. Phys. Rev. E 53 R5584–R5587.
  • Gotcheva, V. and Teitel, S. (2001). Depinning transition of a two-dimensional vortex lattice in a commensurate periodic potential. Phys. Rev. Lett. 86 2126–2129.
  • den Hollander, F. (2000). Large Deviations. Amer. Math. Soc., Providence, RI.
  • Isozaki, Y. and Yoshida, N. (2001). Weakly pinned random walk on the wall: Pathwise descriptions of the phase transition. Stochastic Process. Appl. 96 261–284.
  • Madras, N. and Whittington, S. G. (2002). Self-averaging in finite random copolymers. J. Phys. A Math. Gen. 35 L427–L431.
  • Naidenov, A. and Nechaev, S. (2001). Adsorption of a random heteropolymer at a potential well revisited: Location of transition point and design of sequences. J. Phys. A Math. Gen. 34 5625–5634.
  • Nechaev, S. and Zhang, Y.-C. (1995). Exact solution of the 2D wetting problem in a periodic potential. Phys. Rev. Lett. 74 1815–1818.
  • Nelson, D. R. and Vinokur, V. M. (1993). Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48 13060–13097.
  • Orlandini, E., Tesi, M. C. and Whittington, S. G. (1999). A self-avoiding walk model of random copolymer adsorption. J. Phys. A Math. Gen. 32 469–477.
  • Orlandini, E., Rechnitzer, A. and Whittington, S. G. (2002). Random copolymers and the Morita approximation: Polymer adsorption and polymer localization. J. Phys. A Math. Gen. 35 7729–7751.
  • Privman, V., Forgacs, G. and Frisch, H. L. (1988). New solvable model of polymer-chain adsorption at a surface. Phys. Rev. B 37 9897–9900.
  • Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 403–434.
  • Tang, L.-H. and Chaté, H. (2001). Rare-event induced binding transition of heteropolymers. Phys. Rev. Lett. 86 830–833.
  • van Rensburg, J. (2001). Trees at an interface. J. Statist. Phys. 102 1177–1209.
  • Whittington, S. G. (1998). A directed-walk model of copolymer adsorption. J. Phys. A Math. Gen. 31 8797–8803.