Quantitative homogenization and fluctuations

Stationarity and ergodicity allow arbitrarily slow averaging, so they do not imply a universal algebraic error bound. A much stronger example is finite-range dependence: the sigma fields generated by coefficients in two regions are independent whenever their separation exceeds a fixed range. Together with uniform ellipticity and boundedness, such hypotheses support quantitative stochastic homogenization and large-scale regularity results. Other theories use specified functional inequalities or quantified mixing assumptions. These conditions are alternatives with their own consequences, not interchangeable consequences of ergodicity.

Even after a dependence assumption is fixed, the target matters. An averaged flux, a finite-volume effective tensor, a solution in L², and a pointwise gradient can have different rates and dimensional corrections. Boundary conditions and coefficient regularity also affect estimates. Decay of one pairwise covariance does not automatically control dependence of nonlinear local functions or prove the functional inequalities a theorem uses. Before importing a rate, state exactly which random observable is estimated and which assumptions of the cited theorem apply to it.

For independent cells with conductivities one and four, let ri be the cell resistance and let r̄N be its sample mean. The finite-specimen conductivity with a unit macroscopic slope is aN=1/r̄N. Resistance averages linearly, so its expectation and variance are easy to compute. Conductivity is a nonlinear transformation: its expectation is generally not the reciprocal of expected resistance. Convexity of the reciprocal on positive numbers gives an upward finite-sample bias. Centering a confidence calculation at the limiting effective value therefore requires attention to bias as well as variance.

The classical independent central limit theorem applies to resistance because it is bounded with nonzero variance. Expanding the reciprocal at the limiting mean yields the conductivity central limit law by the delta method. This asymptotic statement describes the distribution after multiplication by √N as N tends to infinity. It is not an exact Gaussian distribution for a small specimen, and its asymptotic variance is not the exact finite-N variance. Quantitative remainder estimates need their own bounds, although the one-dimensional variables here are exceptionally well controlled.

In computations, the sample volume, boundary treatment, discretization, and number of independent realizations all influence uncertainty. Repeating a biased finite-volume estimator reduces its sampling variance around its own mean; it does not automatically remove its boundary bias. Enlarging a domain can reduce both bias and fluctuations, but with different scalings. A useful error budget keeps the discretization error, finite-volume systematic error, and random statistical error separate, rather than treating a single empirical standard deviation as the full error certificate.

The exact one-dimensional calculation teaches a transferable discipline rather than a universal exponent. First identify the random quantity that averages linearly. Next understand how the desired response transforms it. Finally check whether the dependence assumptions justify a law of large numbers, a central limit theorem, or a quantitative tail estimate. In multidimensional random PDEs the corresponding response can involve global correctors and long-range correlations even when the underlying coefficients have short-range dependence. That is why fluctuation theory requires more than counting independent microscopic cells.