Logarithmic Variance for the Height Function of Square-Ice

In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with a=b=c=1 on the square lattice), or, equivalently, of the uniform random homomorphisms from ℤ^2 to ℤ , has logarithmic variance. This establishes a strong form of roughness of this height function.