Abstract. We consider models given by Hamiltonians of the form H ( I;';p;q;t ; " ) = h ( I )+ n X j =1 1 2 p 2 j + V j ( q j ) + "Q ( I;';p;q;t ; " ) where I 2I R d ;' 2 T d , p;q 2 R n , t 2 T 1 . These are higher di- mensional analogues, both in the center and hyperbolic directions, of the models studied in [DLS03, DLS06a, GL06a, GL06b]. All these models present the large gap problem . We show that, for 0 < " 1, under regularity and explicit non- degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O (1). This domain includes resonance lines and, hence, large gaps among d -dimensional KAM tori. The method of proof follows closely the strategy of [DLS03, DLS06a]. The main new phenomenon that appears when the di- mension d of the center directions is larger than one, is the exis- tence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I , they can be contoured. This corresponds to the mechanism called di usion across resonances in the Physics literature. The present paper, however, di ers substantially from [DLS03, DLS06a]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS08], not avail- able when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the res- onances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS03, DLS06a]