Optimal local laws and CLT for the circular Riesz gas
arXiv:2112.05881
Abstract
We study the circular Riesz gas, a one-dimensional system of particles with long-range power-law repulsion. We establish near-optimal rigidity estimates for the gaps at all scales. We then prove a quantitative central limit theorem for linear statistics, combining Stein's method with a mean-field transport. The main difficulty is to control the fluctuations of the local error terms left by the transport, which we do using the rigidity estimates, convexity and monotonicity. In particular, a comparison principle for the Helffer--Sjöstrand equation allows us to treat test functions with low regularity, including indicators of intervals.
arXiv comment51 pages. Lean 4 formalization available at https://github.com/jboursier/circular-riesz-gas-lean