  We prove that a solution of an elliptic operator with periodic coefficients
behaves on large scales like an analytic function, in the sense of
approximation by polynomials with periodic corrections. Equivalently, the
constants in the large-scale $C^{k,1}$ estimate scale exponentially in $k$,
just as for the classical estimate for harmonic functions. As a consequence, we
characterize entire solutions of periodic, uniformly elliptic equations which
exhibit growth like $O(\exp(\delta|x|))$ for small~$\delta>0$. The large-scale
analyticity also implies quantitative unique continuation results, namely a
three-ball theorem with an optimal error term as well as a proof of the
nonexistence of $L^2$ eigenfunctions at the bottom of the spectrum.
