Weighted second-order Poincaré inequalities: Application to RSA models

Mitia Duerinckx (1, 2, 3) , Antoine Gloria (2)

1 ULB - Département de Mathématique [Bruxelles]
2 MEPHYSTO - Quantitative methods for stochastic models in physics
3 LJLL - Laboratoire Jacques-Louis Lions

Consider an ergodic stationary random field A on the ambient space R d. In a recent work we introduced the notion of weighted (first-order) functional inequalities , which extend standard functional inequalities like spectral gap, covariance, and logarithmic Sobolev inequalities, while still ensuring strong concentration properties. We also developed a constructive approach to these weighted inequalities, proving their validity for prototypical examples like Gaussian fields with arbitrary covariance function , Voronoi and Delaunay tessellations of Poisson point sets, and random sequential adsorption (RSA) models, which do not satisfy standard functional inequalities. In the present contribution, we turn to second-order Poincaré inequalities à la Chatterjee: while first-order inequalities quantify the distance to constants for nonlinear functions X(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 weighted second-order Poincaré inequalities. Applied to RSA models, these functional inequalities allow us 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.

Domaines: Probabilités [math.PR]

Mitia Duerinckx, Antoine Gloria. Weighted second-order Poincaré inequalities: Application to RSA models. 2017. ⟨hal-01633042⟩