  Standard statistical mechanical or condensed matter arguments tell us that
bulk properties of a physical system do not depend too much on boundary
conditions. Random tilings of large regions provide counterexamples to such
intuition, as illustrated by the famous 'arctic circle theorem' for dimer
coverings in two dimensions. In these notes, I discuss such examples in the
context of critical phenomena, and their relation to 1+1d quantum particle
models. All those turn out to share a common feature: they are inhomogeneous,
in the sense that local densities now depend on position in the bulk. I explain
how such problems may be understood using variational (or hydrodynamic)
arguments, how to treat long range correlations, and how non trivial edge
behavior can occur. While all this is done on the example of the dimer model,
the results presented here have much greater generality. In that sense the
dimer model serves as an opportunity to discuss broader methods and results.
[These notes require only a basic knowledge of statistical mechanics.]
