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arXiv · 2608.06172

Approximating spin systems on planar graphs

Heng Guo·Xinyuan Zhang
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Abstract

We show that the hard-core partition function admits a fully polynomial-time randomised approximation scheme (FPRAS) on planar graphs when the activity is a sufficiently small constant. In contrast, we show that for any constant $q\ge 4$, approximately counting $q$-colourings in planar graphs is NP-hard. We also give a complete characterisation of when an FPRAS exists for a sufficiently small external field for 2-spin systems on planar graphs. The main ideas of all proofs were found using GPT-5.6 Sol Ultra.

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Heng Guo, Xinyuan Zhang. 2026-08-06. Approximating spin systems on planar graphs. https://arxiv.org/abs/2608.06172

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