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We consider a simple random walk of length $N$, denoted by $(S_{i})_{i\in\{1,...,N\}}$, and we define $(w_i)_{i\geq 1}$ a sequence of centered i.i.d.random variables. For $K\in\N$ we define$((\gamma_i^{-K},...,\gamma_i^K))_{i\geq 1}$ an i.i.d sequence of randomvectors. We set $\beta\in \mathbb{R}$, $\lambda\geq 0$ and $h\geq 0$, andtransform the measure on the set of random walk trajectories with theHamiltonian $\lambda \sum_{i=1}^{N} (w_i+h) \sign(S_i)+\beta\sum_{j=-K}^{K}\sum_{i=1}^{N} \gamma_{i}^{j} \boldsymbol{1}_{\{S_{i}=j\}}$.This transformed path measure describes an hydrophobic(philic) copolymerinteracting with a layer of width $2K$ around an interface between oil andwater.  In the present article we prove the convergence in the limit of weak coupling(when $\lambda$, $h$ and $\beta$ tend to 0) of this discrete model towards itscontinuous counterpart. To that aim we further develop a technique of coarsegraining introduced by Bolthausen and den Hollander in \cite{BDH}. Our resultshows, in particular, that the randomness of the pinning around the interfacevanishes as the coupling becomes weaker.

Copolymer at selective interfaces and pinning potentials: Weak coupling limits