Average Field Approximation for Almost Bosonic Anyons in a Magnetic Field

We study the ground state of a large number N of 2D anyons in an external magnetic field. We consider a scaling limit where the statistics parameter $\alpha$ tends to zero when N tends to infinity which allows the statistics to be seen as a "perturbation from the bosonic end". Our model is that of bosons in a magnetic field and interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows to take R tends to not too fast at the same time as N tends to infinity. We use the information theoretic version of the de Finetti theorem of Brand{\~a}o and Harrow to justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.