  Let $f$ be a Laurent polynomial in $d$ commuting variables with integer
coefficients. Associated to $f$ is the principal algebraic
$\mathbb{Z}^d$-action $\alpha_f$ on a compact subgroup $X_f$ of
$\mathbb{T}^{\mathbb{Z}^d}$ determined by $f$. Let $N\ge1$ and restrict points
in $X_f$ to coordinates in $N\mathbb{Z}^d$. The resulting algebraic
$N\mathbb{Z}^d$-action is again principal, and is associated to a polynomial
$g_N$ whose support grows with $N$ and whose coefficients grow exponentially
with $N$. We prove that by suitably renormalizing these decimations we can
identify a limiting behavior given by a continuous concave function on the
Newton polytope of $f$, and show that this decimation limit is the negative of
the Legendre dual of the Ronkin function of $f$. In certain cases with two
variables, the decimation limit coincides with the surface tension of random
surfaces related to dimer models, but the statistical physics methods used to
prove this are quite different and depend on special properties of the
polynomial.
