Decimation limits of principal algebraic ℤ^d-actions

Let f be a Laurent polynomial in d commuting variables with integer coefficients. Associated to f is the principal algebraic ℤ^d-action α_f on a compact subgroup X_f of T^Zd determined by f. Let N ⩾ 1 and restrict points in X_f to coordinates in Nℤ^d. The resulting algebraic Nℤ^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.