Two dimensional inhomogeneous classical systems at criticality

We study simple inhomogeneous deformations of known two-dimensional classical lattice models described by conformal field theory (CFT) at large distances. The deformations are chosen to vary slowly at lattice scales, while preserving critical behavior. Globally, we find that such systems are described by a CFT in curved space, and identify the underlying space metric. Our two examples are the Ising model and the six vertex model with domain wall boundary conditions. In the latter boundaries are also inhomogeneous, which complicates the analysis. We nevertheless solve the free case using hydrodynamics. In the presence of interactions we determine the arctic curve which separates a critical fluctuating region from an ordered (frozen) phase.