The Algebraicity Problem for Hard-Core Entropy Constants on the Discrete Hypertori

Yotam Svoray

paper

The Algebraicity Problem for Hard-Core Entropy Constants on the Discrete Hypertori

arXiv:2608.28332

Abstract

We use tools and techniques from $p$ -adic analysis and algebraic number theory to study the algebraicity of the hard square entropy constant and its high dimensional analogues. Specifically, we study arithmetic properties of $a_{d} (n)$ , the number of independent sets in the $d$ -dimensional discrete torus, and the associated entropy constants $κ_{d} = lim_{n \to \infty} a_{d} (n)^{1/ n^{d}}$ . It is not known whether $κ_{d}$ is algebraic or transcendental for $d > 1$ . Using the fact that the sequence $a_{d} (p^{k})$ converges $p$ -adically for every prime $p$ and other arithmetic facts, we present a collection of criteria for the algebraicity of $κ_{d}$ and bound the number of possible values of prime powers $p^{k}$ for which $a_{d} (p^{k}) = κ_{d}^{p^{k d}}$ .

arXiv comment33 pages, comments are welcome