Approximation and Hardness Results for Spin Systems on Planar Graphs
Heng Guo and Xinyuan Zhang give an FPRAS for the planar hard-core model at small activity, prove NP-hardness of approximately counting planar $q$-colorings for every $q\geq 4$, and classify small-field two-spin systems; they report that GPT-5.6 Sol Ultra found the main ideas behind all proofs.
In a preprint first submitted on August 6, 2026, Heng Guo and Xinyuan Zhang establish several approximation and hardness results for spin systems on planar graphs. They give an FPRAS for the hard-core partition function at sufficiently small constant activity; prove that, for every , approximately counting proper -colorings of planar graphs is NP-hard, confirming a conjecture of Welsh; and characterize which two-spin systems admit an FPRAS when the external field is sufficiently small. The revised version also notes that a key correlation bound resolves an open problem of Leake and Oveis Gharan.
The authors state that GPT-5.6 Sol Ultra found the main ideas behind all proofs. For the approximation algorithm, the argument combines star-coloring blocks with rapidly mixing block dynamics; for the hardness results, the model helped identify suitable source problems for reductions. Guo and Zhang report that they simplified and streamlined the arguments, wrote all proofs themselves, and take full responsibility for the results.