Report
Decimation limits of principal algebraic $\mathbb{Z}^d$-actions
| العنوان: | Decimation limits of principal algebraic $\mathbb{Z}^d$-actions |
|---|---|
| المؤلفون: | Arzhakova, Elizaveta, Lind, Douglas, Schmidt, Klaus, Verbitskiy, Evgeny |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Dynamical Systems, 37A15, 37A35, 37A44, 13F20 |
| الوصف: | 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. Comment: 33 pages with 8 figures. An informal account, with background and motivation, in contained in a talk by Lind that can be viewed at https://www.youtube.com/watch?v=jT2OQifgldk&t=120s |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2104.04408 |
| رقم الانضمام: | edsarx.2104.04408 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |