Average-field approximation for dilute almost-bosonic anyons

We study the ground state of a system of N two-dimensional trapped almost-bosonic anyons. This setup can equivalently be viewed as bosons interacting through long-range magnetic potentials generated by magnetic charges carried by each particle. These magnetic charges are assumed to be smeared over discs of radius R-a model known as extended anyons. To recover the point-like anyons perspective, we consider the joint limit $R\to 0$ as $N\to \infty$ . We rigorously justify the average-field approximation for any radius R that decays polynomially, and even for certain exponentially decaying R. The average-field approximation asserts that the particles behave like independent, identical bosons interacting through a self-consistent magnetic field. Our result significantly improves upon the best-known estimates from [1, 2], and it in particular covers radii that are much smaller than the mean interparticle distance. The proof strategy builds on a recent work on two-dimensional attractive Bose gases by Junge and the author [3].