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Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK)points with exactly known critical ground states and deconfined spinons. Weexamine generic, weak, perturbations around these points. In d=2+1 we find afirst order transition between a ``plaquette'' valence bond crystal and aregion with a devil's staircase of commensurate and incommensurate valence bondcrystals. In the part of the phase diagram where the staircase is incomplete,the incommensurate states exhibit a gapless photon and deconfined spinons on aset of finite measure, almost but not quite a deconfined phase in a compactU(1) gauge theory in d=2+1! In d=3+1 we find a continuous transition betweenthe U(1) resonating valence bond (RVB) phase and a deconfined staggered valencebond crystal. In an appendix we comment on analogous phenomena in quantumvertex models, most notably the existence of a continuous transition on thetriangular lattice in d=2+1.Comment: 9 pages; expanded version to appear in Phys. Rev. B; presentation  improve

Bipartite Rokhsar–Kivelson points and Cantor deconfinement