Conjecture on norm-dependent optimality of triangular and square lattices

Let $s>2$. Let $\mathcal{L}_2(1)$ denote the unit-covolume planar lattice space and $\mathcal{L}_2$ the planar lattice space, let $\zeta_{L,\|\cdot\|_p}(s)$ be the Epstein zeta function associated with the $p$-norm, and let $\textnormal{E}_{\|\cdot\|_p}^{\textnormal{LJ}}(L)$ be the corresponding Lennard–Jones energy. Write $\mathsf{A}_2$ for the triangular lattice, and let $\lambda_{\mathsf{A}_2}$ and $\lambda_{\mathbb Z^2}$ be the specified optimal scaling factors.

Optimality conjecture for triangular and square lattices. There exist $1<p_1<p_2$ and $1<p_3<p_4$ such that, up to rotation, the following hold:

1. For $L\mapsto\zeta_{L,\|\cdot\|_p}(s)$ in $\mathcal{L}_2(1)$:

• for all $p\in[1,2]$, the minimizer is $\sqrt{\frac{2}{\sqrt{3}}}\mathsf{A}_2$;
• for all $p\in(p_1,p_2)$, the minimizer is $\mathbb Z^2$.

2. For $L\mapsto\textnormal{E}_{\|\cdot\|_p}^{\textnormal{LJ}}(L)$ in $\mathcal{L}_2$:

• for all $p\in(p_3,2]$, the minimizer is $\lambda_{\mathsf{A}_2}\mathsf{A}_2$;
• for all $p\in\{1\}\cup(p_4,\infty]$, the minimizer is $\lambda_{\mathbb Z^2}\mathbb Z^2$.

The conjecture predicts phase transitions in the optimal lattice as the norm varies. The triangular lattice is optimal in the stated low-$p$ Epstein-zeta range, while square-lattice optimality occurs on an intermediate range; for the Lennard–Jones energy, the triangular and square lattices are predicted on the stated ranges. The claims are based on numerical simulations and remain open.