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Pinning of polymers and interfaces by random potentials (2005)
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BibTeX
@MISC{Alexander05pinningof,
author = {S. Alexander and Vladas Sidoravicius},
title = {Pinning of polymers and interfaces by random potentials},
year = {2005}
}
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Abstract
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. 1. Introduction. Consider
Keyphrases
random potential critical point self-averaging occurs arbitrary temperature markov chain main result added randomness one-dimensional defect high probability quenched free energy excursion length distribution critical value mean-0 randomness null set directed polymer finite exponential moment interaction vary polynomial tail quenched case deterministic case deterministic interaction monomer location similar situation positive fraction low temperature particular site
