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Abstract: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.

Submission history

From: Douglas Lind [view email]
[v1] Fri, 9 Apr 2021 14:59:05 UTC (20,644 KB)
[v2] Mon, 22 Aug 2022 23:08:37 UTC (20,643 KB)
[v3] Sat, 30 May 2026 19:10:15 UTC (7,370 KB)