  We consider the six-vertex model in an L-shaped domain of the square lattice,
with domain wall boundary conditions. For free-fermion vertex weights the
partition function can be expressed in terms of some Hankel determinant, or
equivalently as a Coulomb gas with discrete measure and a non-polynomial
potential with two hard walls. We use Coulomb gas methods to study the
partition function in the thermodynamic limit. We obtain the free energy of the
six-vertex model as a function of the parameters describing the geometry of the
scaled L-shaped domain. Under variations of these parameters the system
undergoes a third-order phase transition. The result can also be considered in
the context of dimer models, for the perfect matchings of the Aztec diamond
graph with a cut-off corner.
