Multiscale second-order Poincar\'e inequalities in probability

Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In a companion article, we introduced the notion of multiscale (first-order) functional inequalities, which extend standard functional inequalities like Poincar'e, covariance, and logarithmic Sobolev inequalities in the probability space, while still ensuring strong concentration properties. We also developed a constructive approach to these functional inequalities, proving their validity for prototypical examples including Gaussian fields, Poisson random tessellations, and random sequential adsorption (RSA) models, which do not satisfy standard functional inequalities. In the present contribution, we turn to second-order Poincar'e inequalities `a la Chatterjee: while first-order inequalities quantify the distance to constants for nonlinear functions $Z(A)$ in terms of their local dependence on the random field $A$, second-order inequalities quantify their distance to normality. For the above-mentioned examples, we prove the validity of suitable multiscale second-order Poincar'e inequalities. In particular, applied to RSA models, these functional inequalities allow to complete and improve previous results by Schreiber, Penrose, and Yukich on the jamming limit, and to propose and fully analyze a more efficient algorithm to approximate the latter.