Tail estimates for homogenization theorems in random media

Abstract

Abstract. Consider a random environment in Z d given by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environments and the spectral gap of the random walk in a finite cube.Mathematics Subject Classification. 60K37, 35B27, 82B44.

A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts

“…Other noteworthy earlier derivations of (suboptimal) variance upper bounds include an old paper by Yurinskii [24] and a more recent paper by Benjamini and Rossignol [2]. Closely related to these estimates are recent derivations of quantitative central limit theorems for random walk among random conductances and approximations of the limiting diffusivity matrix, e.g., Caputo and Ioffe [7], Bourgeat and Piatnitski [5], Boivin [4], Mourrat [18], Gloria and Mourrat [9,10], etc. Incidentally, the Meyers estimate is also the key tool in [7].…”

“…Unfortunately, a direct attempt at the substitution of (the gradients of) Ψ Λ by ψ in (2.13) reveals another technical obstacle: As (2.13) relies on the Fundamental Theorem of Calculus, the replacement of Ψ Λ by ψ requires the latter function to be defined for ω that may lie outside of the support of P. This is a problem because ψ is generally determined by conditions (1)(2)(3)(4) in Proposition 2.2 only on a set of full P-measure. Imposing additional assumptions on P -namely, that the single-conductance distribution is supported on an interval with a bounded and non-vanishing density -would allow us to replace the Lebesgue integral in (2.13) by an integral with respect to P(dω k ) and thus eliminate this problem.…”

Noise-stability and central limit theorems for effective resistance of random electric networks

“…In [5], the definition of A n is different, based on the discrete cube and not on the torus, however the behaviour should be the same. [5] obtains only suboptimal bounds for the variance of the mean conductivity.…”

“…The reciprocity law (5) gives that i e r (e ′ ) and i e ′ r (e) are of the same order: but e ′ r(e ′ )(i e r (e ′ )) 2 is the effective resistance from e − to e + , which is of order at most 1 [in fact at most r(e)]. Thus,…”

“…In [5], the definition of A n is different, based on the discrete cube and not on the torus, however the behaviour should be the same. [5] obtains only suboptimal bounds for the variance of the mean conductivity. In [16], an optimal bound on the variance is obtained for a related quantity based on the corrector of homogenization theory.…”