Centre d'Etudes de Saclay, 91191 Gif sur Yvette cedex, France

A basic question in quantum many body theory is to know how one goes from independent particle motion towards collective motion when one decreases the density of a system of charged particles repelling each other via a Coulomb repulsion. The motivation to re-visit nowadays this question comes from the possibility to create two dimensional gases of charges in high quality field effect devices and to decrease by a gate the carrier density down to a very dilute limit. Conductance measurements for different densities as a function of the temperature, of the bias voltage, of a parallel magnetic field, etc, show the existence of an unexpected low temperature metallic behavior when one goes towards the dilute limit. This raises the question of the existence of an intermediate metallic phase between two insulating phases of different nature: the Fermi glass of Anderson localized states at large and the pinned Wigner solid at low . This question is numerically investigated using mesoscopic lattice models with and without disorder.

In **lecture 1**, the recent experimental results motivating to re-visit
the Fermi-Wigner crossover will be reviewed.

In **lecture 2**, detailed exact numerical studies of a few electron
mesoscopic lattice models will be presented, showing a specific
intermediate regime between the weak and strong coupling limits.

In **lecture 3**, the conjecture of an intermediate phase between the
Fermi liquid and the Wigner solid, first proposed by Andreev-Lifshitz,
will be discussed in relation with the numerical results.

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