Localization transition for a polymer near an interface

• 成果类型: Article
• 署名作者: Bolthausen, E; den Hollander, F
• 署名单位: University of Zurich; Radboud University Nijmegen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1334-1366
• 关键词: partial sums; random-walk; approximation

摘要

Consider the directed process (i, S-i)(i greater than or equal to 0) where the second component is simple random walk on Z (S-0 = 0). Define a transformed path measure by weighting each n-step path with a factor exp[lambda Sigma(1 less than or equal to i less than or equal to n) (omega(i) + h)sign(S-i)]. Here, (omega(i))(i greater than or equal to 1) is an i.i.d. sequence of random variables taking values +/- 1 with probability 1/2 lading as a random medium), while lambda is an element of [0, infinity) and h is an element of [0, 1) are parameters. The weight factor has a tendency to pull the path towards the horizontal, because it favors the combinations S-i > 0, omega(i) = +1 and S-i < 0, omega(i) = -1. The transformed path measure describes a heteropolymer, consisting of hydrophylic and hydrophobic monomers, near an oil-water interface. We study the free energy of this model as n --> infinity and show that there is a critical curve lambda --> h(c)(lambda) where a phase transition occurs between localized and delocalized behavior tin the vertical direction). We derive several properties of this curve, in particular, its behavior for lambda down arrow 0. To obtain this behavior, we prove that as lambda, h down arrow 0 the free energy scales to its Brownian motion analogue.