Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz’s theorem

We investigate a very general technique to obtain CLTs with near-optimal rates of convergence for broad classes of strongly dependent stochastic systems, based on the zeros of the characteristic function. Using this, we demonstrate Gaussian fluctuations for the magnetization (i.e., the total spin) for a large class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins, at the near-optimal rate of $O(\log |\Lambda|\cdot|\Lambda|^{-1/2})$ for system size $|\Lambda|$. This includes, in particular, the celebrated $XY$ and Heisenberg models under ferromagnetic conditions.
Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case a technique of Lebowitz, Ruelle, Pittel and Speer for deriving CLTs in discrete statistical mechanical models, for which we obtain sharper convergence rates. In a very different application, we obtain CLTs for linear statistics of a wide class of point processes known as $\alpha$-determinantal processes which interpolate between negatively and positively associated random point fields (including the usual determinantal, permanental and Poisson processes). Notably, we address strongly correlated processes in dimensions $\ge\>3$, where connections to random matrix theory are not available, and handle a broad class of kernels including those with slow spatial decay (such as the Bessel kernel in general dimensions). A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that we establish under the significantly milder condition that the characteristic function is non-vanishing only on a bounded disk. Joint work with T.C. Dinh, H.S. Tran and M.H. Tran.