In this paper, we consider Schrödinger operators on M×Zd2$M\times {\mathbb{Z}}^{d_2}$, with M={M1,…,M2}d1$M=\{M_1,\dots ,M_2{\}}^{d_1}$ (‘quantum wave guides’) with a ‘Γ$\mathrm{\Gamma }$-trimmed’ random potential, namely a potential which vanishes outside a subset Γ$\mathrm{\Gamma }$ which is periodic with respect to a sub-lattice. We prove that (under appropriate assumptions) for strong disorder these operators have pure point spectrum outside the set Σ0=σ(H0,Γc)${\text{Σ}}_0=\text{σ}(H_{0,{\mathrm{\Gamma }}^c})$ where H0,Γc$H_{0,{\mathrm{\Gamma }}^c}$ is the free (discrete) Laplacian on the complement Γc${\mathrm{\Gamma }}^c$ of Γ$\mathrm{\Gamma }$. We also prove that the operators have some absolutely continuous spectrum in an energy region E⊂Σ0$\mathcal{E}\subset {\text{Σ}}_0$. Consequently, there is a mobility edge for such models. We also consider the case −M1=M2=∞$-M_1=M_2=\mathrm{\infty }$, i.e. Γ$\mathrm{\Gamma }$-trimmed operators on Zd=Zd1×Zd2${\mathbb{Z}}^d={\mathbb{Z}}^{d_1}\times {\mathbb{Z}}^{d_2}$. Again, we prove localisation outside Σ0${\text{Σ}}_0$ by showing exponential decay of the Green function GE+iη(x,y)$G_{E+i\eta }(x,y)$ uniformly in η>0$\eta >0$. For all energies E∈E$E\in \mathcal{E}$ we prove that the Green’s function GE+iη$G_{E+i\eta }$ is not (uniformly) in ℓ1${\ell }^1$ as η$\eta$ approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.