Decimation limits of principal algebraic $\mathbb{Z}^d$-actions

TL;DR

This paper investigates the asymptotic behavior of principal algebraic $Z^d$-actions associated with Laurent polynomials, revealing a limiting function related to the Ronkin function and connecting to concepts like surface tension in statistical physics.

Contribution

It establishes a connection between decimation limits of algebraic actions and the Legendre dual of the Ronkin function, providing a new analytical framework for understanding these systems.

Findings

Decimation limits form a continuous concave function on the Newton polytope.

The decimation limit equals the negative Legendre dual of the Ronkin function.

In two-variable cases, the limit matches the surface tension of related dimer models.

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.