Geometric bounds for spanning tree entropy of planar lattice graphs

TL;DR

The paper investigates the spanning-tree entropy $z_{\mathcal{G}}$ of planar lattice graphs by linking combinatorial growth to hyperbolic geometry via associated biperiodic links. It introduces geometric invariants such as the bipyramid volume $\nu^{\lozenge}(\mathcal{G})$ and the right-angled volume $vol^{\perp}(\mathcal{G})$, establishing bounds $vol(\mathcal{G}) \le 2\pi z_{\mathcal{G}} \le \overline{\nu}(\mathcal{G})$ with $0<vol(\mathcal{G})$, and demonstrates, for many graphs, that $z_{\mathcal{G}}$ can be tightly estimated by these volumes. The authors prove infinitely many cases of the conjectured bound by constructing families via parallel edges, truncation, and medial operations, and they reveal a deep connection with the toroidal dimer model in isoradial embeddings where the lower bound can be achieved with equality for the regular lattices. The work also addresses finite planar graphs and the Vol-Det conjecture, illustrating how hyperbolic volumes bound spanning-tree counts and how these ideas extend from finite graphs to infinite lattice limits. Overall, the contributions blend graph theory, hyperbolic geometry, and statistical mechanics to bound and often exactly determine spanning-tree entropy in planar lattices, with practical computational routes and notable counterexamples clarifying the scope of the bounds.

Abstract

We prove infinitely many cases of conjectured sharp upper and lower bounds for the spanning tree entropy of any planar lattice graph. These bounds come from volumes of associated hyperbolic alternating links, right-angled hyperbolic polyhedra and hyperbolic regular ideal bipyramids. For many planar lattice graphs, we show these bounds are easy to compute and provide excellent numerical estimates for the spanning tree entropy.

Figures (6)

• Figure 1: (a) Biperiodic triaxial link $\mathcal{L}$, and fundamental domain for $L$. (b) Temperleyan graph $\mathcal{G}^{b}$, and fundamental domain for $G^b$.
• Figure 2: (a) A tetrahedron with stellating (blue), vertical (green), and horizontal (black) edges. A link triangle for an ideal vertex at $\infty$ is shaded. (b) Tetrahedra are glued at the stellating edge, centrally triangulating faces of $G$ on the torus. (c) Shaded triangles indicate the four tetrahedra glued along one horizontal edge. Right: Volumes of hyperbolic regular ideal $n$-bipyramids.
• Figure 3: Some planar lattice graphs in Table \ref{['table1']}. Figures for Lattice graphs #1-5 and 7 are from TeuflWagner. Figures for Lattice graphs #14-16 are from semi-regular-wiki. For Lattice graphs #8-9, see Figure \ref{['fig:dual-list']}.
• Figure 4: Biperiodic alternating link whose Tait graph is Lattice graph #11 in Table \ref{['table1']}.
• Figure 5: Lattice graphs $\mathcal{G}$ and $\mathcal{G}^*$ with orthogonally dual embeddings shown, so faces of $\mathcal{G}^{b}$ are right kites in each case. Labels from Table \ref{['table1']}. Figures from Cairo-wiki.

Theorems & Definitions (28)

• Conjecture 1
• Definition 2
• Proposition 3
• proof
• Definition 4
• Proposition 5
• proof
• Proposition 8
• proof
• Conjecture 9