Logarithmic variance for the height function of square-ice

Hugo Duminil-Copin, Matan Harel, Benoit Laslier, Aran Raoufi, Gourab Ray

Published 2019-10-31Version 1

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 $\mathbb Z^2$ to $\mathbb Z$, has logarithmic variance. This establishes a strong form of roughness of this height function.