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.