  The quantitative analysis of stochastic homogenization problems has been a
very active field in the last fifteen years. Whereas the first results were
motivated by applied questions (namely, the numerical approximation of
homogenized coefficients), the more recent achievements in the field are much
more analytically-driven and focus on the subtle interplay between PDE analysis
(and in particular elliptic regularity theory) and probability (concentration,
stochastic cancellations, scaling limits). The aim of this article is
threefold. First we provide a complete and self-contained analysis for the
popular example of log-normal coefficients with possibly fat tails in dimension
$d=1$, establishing new results on the accuracy of the two-scale expansion and
characterizing fluctuations (in the perspective of uncertainty quantification).
Second, we work in a context where explicit formulas allow us to by-pass
analytical difficulties and therefore mostly focus on the probabilistic side of
the theory. Last, the one-dimensional setting gives intuition on the available
results in higher dimension (provided results are correctly reformulated) to
which we give precise entries to the recent literature.
