  Anyons are 2D or 1D quantum particles with intermediate statistics,
interpolating between bosons and fermions. We study the ground state of a large
number N of 2D anyons, in a scaling limit where the statistics parameter is
proportional to the inverse of N. This means that the statistics is seen as a
"perturbation from the bosonic end." We model this situation in the magnetic
gauge picture by bosons interacting through long-range magnetic potentials. We
assume that these effective statistical gauge potentials are generated by
magnetic charges carried by each particle, smeared over discs of radius R
(extended anyons). Our method allows to take R to 0 not too fast at the same
time as N to infinity. In this limit we rigorously justify the so-called
"average field approximation": the particles behave like independent,
identically distributed bosons interacting via a self-consistent magnetic
field.
