Copolymer at selective interfaces and pinning potentials: Weak coupling limits

We consider a simple random walk of length N, denoted by (S(i))(i is an element of{1,...,N}), and we define (w(i))(i >= 1) a sequence of centered i.i.d. random variables. For K is an element of N we define ((gamma(-K)(i).....gamma(K)(i)))(i >= 1), an i.i.d sequence of random vectors. We set beta is an element of R, lambda >= 0 and h >= 0. and transform the measure on the set of random walk trajectories with the Hamiltonian lambda Sigma(N)(i=1)(w(i) +h) sign (S(i)) + beta Sigma(K)(j=-K)Sigma(N)(i=1) gamma(j)(i) (1){S(i=j)}. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2K around an interface between oil and water. 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 its Continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in Ann. Probab. 25 (1997) 1334-1366. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.