The Annals of Applied Probability

Pinning of polymers and interfaces by random potentials

Kenneth S. Alexander and Vladas Sidoravicius

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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

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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.

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