Localization transition for a polymer near an interface

Abstract

Consider the directed process (i S i ) i 0 where the second component is simple random walk on Z(S 0 = 0). De ne a transformed path measure by w eighting each n-step path with a factor exp P 1 i n (! i +h)sign(S i )]. Here, (! i ) i 1 is an i.i.d. sequence of random variables taking values 1 with probability 1 =2 (acting as a random medium), while 2 0 1) a n d h 2 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 ! i = +1 and S i < 0 ! 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 ! 1 , and show that there is a critical curve ! h c ( ) where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive s e v eral properties of this curve, in particular, its behavior for # 0. To obtain this behavior, we prove that as h # 0 the free energy scales to its Brownian motion analogue.AMS 1991 subject classi cations. 60F10, 60J15, 82B26.

A Numerical Approach to Copolymers at Selective Interfaces

“…(4) Finally, in Section 5, we report the results of a numerical attempt to determine the critical curve. While this issue has to be treated with care, mostly for the reasons raised in point 4 above, we observe a surprising phenomenon: the critical curve appears to be very close to h (m) (·) for a suitable value of m. By the universality result proven in [15], building on the free energy Brownian scaling result proven in [5], the slope at the origin of h c (·) does not depend on the law of ω. Therefore if really h (m) (·) = h c (·), since the slope at the origin of h (m) (·) is m, m is the universal constant we are looking for.…”

“…This universal character of the slope at the origin makes this quantity very interesting. Theorem 1.1 is a mild generalization of the results proven in [5] and [3]: the extension lies in the fact that ω 1 is not necessarily symmetric and a proof of it requires minimal changes with respect to the arguments in [3]. The lower bound in (1.7) is actually proven explicitly in Appendix B (see also Section 3), but we stress that we present this proof because it is a new one and because it gives some insight on the computational results.…”

“…We stress that, even if equivalent to Z N,ω exp(−λhN ) in the Laplace asymptotic sense, Z N,ω turns out to be substantially more useful. This had been already realized in [5], but for our results looking at Z N,ω , rather than Z N,ω , is even more essential. Moreover from now on f(λ, h), rather than f (λ, h), will be for us the free energy.…”

Localization of a random copolymer at an interface: an exact enumeration study

“…Localization of a random copolymer at an interface has been investigated by Sinai and Spohn (1966), Bolthausen and den Hollander (1997), Maritan et al (1999), Biskup and den Hollander (1999), Martin et al (2000) and Madras and Whittington (2003). The model considered here is an extension of that introduced by Martin et al (2000) and has also been investigated by Madras and Whittington (2003).…”

Diffusion of a Heteropolymer in a Multi-Interface Medium

“…\Ve begin by describing the on,,"~interface model that was studied in Bolthausen and den Hollander [3]. This model has two ingredients:…”