New Planar Algorithms and a Full Complexity Classification of the Eight-Vertex Model

TL;DR

This work delivers a complete complexity classification for the planar eight-vertex model across all complex parameters, establishing a trichotomy: (i) tractable on all graphs, (ii) tractable on planar graphs but #P-hard on general graphs, or (iii) #P-hard even for planar graphs. The authors develop a systematic framework based on Holant problems, integrating gadget constructions, holographic transformations, and advanced interpolation techniques. A central result is the discovery of a new planar tractable regime for symmetric weights, obtained by a local combinatorial reduction to the planar Even Coloring problem and a holographic step into a planar six-vertex instance, enriching the known taxonomy beyond the six-vertex planarity results. They also reveal intricate nonlocal connections to the bipartite Ising model, conformal lattice interpolation, and cyclotomic-field methods, highlighting deep links between combinatorial gadgetry, complex-analytic transforms, and statistical physics models. The methods collectively advance the Holant dichotomy program, offering new tools to identify tractable symmetric and asymmetric counting problems on planar graphs with potential extensions to broader Holant settings.

Abstract

We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) \#P-hard for general graphs but computable in P-time for planar graphs, or (3) \#P-hard even for planar graphs. The classification has an explicit criterion. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and Möbius transformation from complex analysis. The proof also makes use of cyclotomic fields.

Figures (8)

• Figure 1: Valid configurations of the eight-vertex model.
• Figure 2: A 4-ary signature $f$ with four edge inputs labeled counterclockwise as $x_1, x_2, x_3, x_4$.
• Figure 3: An $\mathcal{F}$-gate with 5 dangling edges.
• Figure 4: The movement of the entries in the signature matrix of an arity 4 signature under a clockwise rotation of the input edges. The Hamming weight two entries are in the two solid cycles (one has length 4 and the other one is a swap).
• Figure 5: Connect variables $x_\ell$, $x_k$ of $f_1$ with variables $x_s$, $x_t$ of $f_2$ both using $\neq_2$.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• Lemma 2.6
• proof