Copolymer at selective interfaces and pinning potentials: weak coupling limits

We consider a simple random walk of length N, denoted by (Si)i¿{1, …, N}, and we define (wi)i=1 a sequence of centered i.i.d. random variables. For K¿N we define ((¿i-K, …, ¿iK))i=1 an i.i.d sequence of random vectors. We set ß¿R, ¿=0 and h=0, and transform the measure on the set of random walk trajectories with the Hamiltonian ¿¿i=1N(wi+h)sign(Si)+ß¿j=-KK¿i=1N¿ij1{Si=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 ¿, h and ß 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.