Let H be a lattice in a higher-rank semisimple Lie group. It is known that any group quasi-isometric to H is also virtually a lattice in the same Lie group. We investigate similar behavior in the world of right-angled Artin groups (RAAG). More precisely, it is known that a RAAG acts geometrically on a canonical CAT(0) cube complex. Then the question is whether any group H quasi-isometric to this RAAG also virtually acts geometrically on the same CAT(0) cube complex. While the answer is no in general, we show for large classes of RAAGs, H acts geometrically on a CAT(0) cube complex which is a simple deformation of the original CAT(0) cube complex, with aspects of the polyhedron product structure preserved. Based on joint work with B. Kleiner.