We consider a class of singlet resonanting-valence-bond wave functions on a square lattice, with the bond-length distribution as a variational parameter. This class contains the two limiting cases of the dimer wave function and of the Neel state. We present numerical calculations of the energy and the spinspin correlation functions up to very large lattice sizes (180x180) both for disordered states with exponentially decaying correlation functions and for ordered states. The energy of a disordered state can be within 0.1% of our best ordered state ( -0.33447/bond).

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We present an extreme localization mechanism induced by a magnetic field for tight-binding electrons in two-dimensional structures. This spectacular phenomenon is investigated for a large class of tilings (periodic, quasiperiodic, or random). We are led to introduce the Aharonov-Bohm cages defined as the set of sites eventually visited by a wavepacket that can, for particular values of the magnetic flux, be bounded. We finally discuss the quantum dynamics which exhibits an original pulsating behaviour.