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Section: New Results

Weighted functional inequalities

Functional inequalities like spectral gap, covariance, or logarithmic Sobolev inequalities are powerful tools to prove nonlinear concentration of measure properties and central limit theorem scalings. Besides their well-known applications in mathematical physics (e.g. for the study of interacting particle systems like the Ising model or for interface models), such inequalities were recently used by the team to establish quantitative stochastic homogenization results.

These functional inequalities have nevertheless two main limitations for stochastic homogenization. On the one hand, whereas only few examples are known to satisfy them (besides product measures, Gaussian measures, and more general Gibbs measures with nicely behaved Hamiltonians), these inequalities are not robust with respect to various simple constructions: for instance, a Poisson point process satisfies a spectral gap, but the random field corresponding to the Voronoi tessellation of a Poisson point process does not. On the other hand, these functional inequalities require random fields to have an integrable covariance, which prevents one to consider fields with heavier tails.

In the series of work [26], [27], [28], M. Duerinckx and A. Gloria introduced weaker versions of these functional inequalities in the form of weighted inequalities. The interest of such inequalities is twofold: first, as their unweighted counterpart they ensure strong concentration properties; second, they hold for a large class of statistics of interest to homogenization (which is shown using a constructive approach).