  We address the question of the dependence of the bulk free energy on boundary
conditions for the six vertex model. Here we compare the bulk free energy for
periodic and domain wall boundary conditions. Using a determinant
representation for the partition function with domain wall boundary conditions,
we derive Toda differential equations and solve them asymptotically in order to
extract the bulk free energy. We find that it is different and bears no simple
relation with the free energy for periodic boundary conditions. The six vertex
model with domain wall boundary conditions is closely related to algebraic
combinatorics (alternating sign matrices). This implies new results for the
weighted counting for large size alternating sign matrices. Finally we comment
on the interpretation of our results, in particular in connection with domino
tilings (dimers on a square lattice).
