Approximating spin systems on planar graphs

2026-08-06

AI Summary

This work investigates the complexity of approximately counting configurations in two-spin systems—such as the hard-core model and graph coloring—on planar graphs. By integrating computational complexity theory, probabilistic approximation algorithms, and AI-assisted mathematical reasoning (GPT-5.6 Sol Ultra), it establishes for the first time a complete characterization of the necessary and sufficient conditions under which a fully polynomial randomized approximation scheme (FPRAS) exists in the regime of small external fields or small activity parameters. The main contributions include proving the existence of an FPRAS for the hard-core model with small activity parameters and demonstrating that approximate counting of proper $q$-colorings on planar graphs is NP-hard for $q \geq 4$, thereby providing a full criterion for the existence of FPRAS in this class of problems.

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.

Problem

• spin systems
• planar graphs
• approximate counting
• hard-core model
• q-colourings

Innovation

• FPRAS
• planar graphs
• hard-core model
• q-colourings
• 2-spin systems