  We study numerically the "analyticity breakdown" transition in 1-dimensional
quasi-periodic media. This transition corresponds physically to the transition
between pinned down and sliding ground states. Mathematically, it corresponds
to the solutions of a functional equation losing their analyticity properties.
We implemented some recent numerical algorithms that are efficient and backed
up by rigorous results so that we can compute with confidence even close to the
breakdown. We have uncovered several phenomena that we believe deserve a
theoretical explanation: A) The transition happens in a smooth surface. B)
There are scaling relations near breakdown. C) The scaling near breakdown is
very anisotropic. Derivatives in different directions blow up at different
rates. Similar phenomena seem to happen in other KAM problems.
