Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with a factor $\exp [\lambda \sum_{1 \leq i \leq n}(\omega_i + h)\sign (S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an i.i.d. sequence of random variables taking values $\pm 1$ with probability 1/2 (acting as a random medium) , while $\lambda \in [0, \infty)$ and $h \in [0, 1)$ 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 $n \to \infty$ and show that there is a critical curve $\lambda \to h_c (\lambda)$ where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive several properties of this curve, in particular, its behavior for $\lambda \downarrow 0$. To obtain this behavior, we prove that as $\lambda, h \downarrow 0$ the free energy scales to its Brownian motion analogue.

Bolthausen, E; den Hollander, F (1997). *Localization transition for a polymer near an interface.* The Annals of Probability, 25(3):1334-1366.

## Abstract

Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with a factor $\exp [\lambda \sum_{1 \leq i \leq n}(\omega_i + h)\sign (S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an i.i.d. sequence of random variables taking values $\pm 1$ with probability 1/2 (acting as a random medium) , while $\lambda \in [0, \infty)$ and $h \in [0, 1)$ 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 $n \to \infty$ and show that there is a critical curve $\lambda \to h_c (\lambda)$ where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive several properties of this curve, in particular, its behavior for $\lambda \downarrow 0$. To obtain this behavior, we prove that as $\lambda, h \downarrow 0$ the free energy scales to its Brownian motion analogue.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | andom walk; Brownian motion; random medium; large deviations; phase transition |

Language: | English |

Date: | 1997 |

Deposited On: | 30 Apr 2010 12:23 |

Last Modified: | 05 Apr 2016 13:26 |

Publisher: | Institute of Mathematical Statistics |

ISSN: | 0091-1798 |

Publisher DOI: | https://doi.org/10.1214/aop/1024404516 |

Official URL: | http://projecteuclid.org/euclid.aop/1024404516 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1457622 http://www.zentralblatt-math.org/NEW/zmath/en/search/?q=an%3A0885.60022 |

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