# Logarithmic variance for the height function of square-ice

Harel Matan
Raoufi Aran
Ray Gourab
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Abstract:

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.

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