Average-field approximation for very dilute almost-bosonic anyon gases

We study the ground state of a system of $N$ two-dimensional trapped almost-bosonic anyons subject to an external magnetic field. 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 $R \rightarrow 0$ as $N \rightarrow \infty$. We rigorously justify the average-field approximation for any radii $R$ that decay polynomially in $1/N$, 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 by Lundholm-Rougerie (2015) and Girardot (2020), 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 (2025).