On homogenization and scaling limit of some gradient perturbations of a massless free field

Abstract

We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models.

The additive structure of elliptic homogenization

“…Similarly, (11.5) was shown to hold jointly over e, see [30,Remark 1.4]. In these works, the space is the discrete lattice Z d , d ≥ 3, and a key assumption is that the probability measure has an underlying product structure which makes available tools such as concentration inequalities, the Chatterjee-Stein [9,10] method of normal approximation and the Helffer-Sjöstrand representation of correlations [26,39,32]. Each of these papers makes essential use of, and refines, the optimal quantitative estimates first proved in [19,20,16].…”

“…It is only very recently that satisfactory progress has been made in overcoming this basic obstacle, and by now there are essentially two alternative programs. The first has its origins in an unpublished paper of Naddaf and Spencer [33], and is based on probabilistic machinery more commonly used in statistical physics [32], namely concentration inequalities, such as spectral gap or logarithmic Sobolev inequalities, which provide a way to quantitatively measure the dependence of the solutions on the coefficients. This approach has been developed extensively by Gloria, Otto and their collaborators [19,20,23,16,15,28], who proved optimal quantitative bounds on the scaling of the first-order correctors (including their sublinear growth and spatial averages of their energy density).…”

Gaussian free fields for mathematicians

“…(To be precise, [Ken01] studies random discrete height functions h ǫ -for which the lattice spacing is ǫ-and shows that for smooth density functions ρ, (h ǫ , ρ) converges in law to (h, ρ) = (h, −∆ −1 ρ) ∇ .) Also, [NS97,GOS01] give a similar Gaussian free field convergence result for a class of discretized random surfaces known as Ginzburg-Landau ∇φ random surfaces or anharmonic crystals and [GOS01] shows further that certain time-varying versions of these processes converge to the dynamic GFF.…”

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

“…Hence, also in the scalar case both conditions (25) and (28) cannot be true for every coefficient field a ∈ ( [28], Example 3). (iii) For α ∈ (0, 1), even a suboptimal assumption on the decay of (28) aŝ…”

“…In the case of the dipole gas this implies that the density of the gas must be extremely small, and with no reasonable estimate on how large the density is allowed to be. In [25] Naddaf and Spencer pioneered an alternative approach to estimating correlation functions for the dipole gas which was based on convexity theory. Their starting point was the observation that a correlation function closely related to the charge-charge correlation function for the dipole gas is equal to the integral over time of an averaged Green function for a parabolic PDE in divergence form with random coefficients.…”