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Zero-Freeness Is All You Need: A Weitz-Type FPTAS for the Entire Lee-Yang Zero-Free Region

Authors Shuai Shao , Ke Shi

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LIPIcs.ITCS.2026.114.pdf
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ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
Keywords
  • Ferromagnetic Ising Model
  • Lee–Yang Theorem
  • Weitz-Type FPTAS
  • Strong Spatial Mixing
  • Random Cluster Model

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Abstract

We present a Weitz-type FPTAS for the ferromagnetic Ising model across the entire Lee–Yang zero-free region, without relying on the strong spatial mixing (SSM) property. Our algorithm is Weitz-type for two reasons. First, it expresses the partition function as a telescoping product of ratios, with the key being to approximate each ratio. Second, it uses Weitz’s self-avoiding walk tree, and truncates it at logarithmic depth to give a good and efficient approximation. The key difference from the standard Weitz algorithm is that we approximate a carefully designed edge-deletion ratio instead of the marginal probability of a vertex being assigned a particular spin, ensuring our algorithm does not require SSM.
Furthermore, by establishing local dependence of coefficients (LDC), we indeed prove a novel form of SSM for these edge-deletion ratios, which, in turn, implies the standard SSM for the random cluster model. This is the first SSM result for the random cluster model on general graphs, beyond lattices. Our proof of LDC is based on a new division relation, and we show such relations hold quite universally. This leads to a broadly applicable framework for proving LDC across a variety of models, including the Potts model, the hypergraph independence polynomial, and Holant problems. Combined with existing zero-freeness results for these models, we derive new SSM results for them.

Cite As Get BibTex

Shuai Shao and Ke Shi. Zero-Freeness Is All You Need: A Weitz-Type FPTAS for the Entire Lee-Yang Zero-Free Region. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 114:1-114:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.114

Author Details

Shuai Shao
  • School of Computer Science and Technology & Hefei National Laboratory, University of Science and Technology of China, China
Ke Shi
  • School of Computer Science and Technology & Hefei National Laboratory, University of Science and Technology of China, China

Funding

Supported by the Quantum Science and Technology - National Science and Technology Major Project (QNMP), 2021ZD0302901.

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful comments.

References

  1. Antar Bandyopadhyay and David Gamarnik. Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures & Algorithms, 33(4):452-479, 2008. URL: https://doi.org/10.1002/rsa.20236.
  2. Alexander Barvinok. Combinatorics and complexity of partition functions, volume 30. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-51829-9.
  3. A.G. Basuev. Ising model in half-space: A series of phase transitions in low magnetic fields. Theor. Math. Phys., 153:1539-1574, 2007. Google Scholar
  4. Ferenc Bencs, Khallil Berrekkal, and Guus Regts. Deterministic approximate counting of colorings with fewer than 2 $1delta colors via absence of zeros. arXiv preprint arXiv:2408.04727, 2024. Google Scholar
  5. Ferenc Bencs and Pjotr Buys. Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs. Random Structures & Algorithms, 66(4):e70018, 2025. URL: https://doi.org/10.1002/rsa.70018.
  6. Ferenc Bencs, Péter Csikvári, Piyush Srivastava, and Jan Vondrák. On complex roots of the independence polynomial. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 675-699. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch29.
  7. Ferenc Bencs and Guus Regts. Barvinok’s interpolation method meets weitz’s correlation decay approach. arXiv preprint arXiv:2507.03135, 2025. URL: https://doi.org/10.48550/arXiv.2507.03135.
  8. Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Stefankovic. Approximation via correlation decay when strong spatial mixing fails. SIAM Journal on Computing, 48(2):279-349, 2019. URL: https://doi.org/10.1137/16M1083906.
  9. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Spectral independence via stability and applications to holant-type problems. TheoretiCS, 3, 2024. URL: https://doi.org/10.46298/theoretics.24.16.
  10. Weiming Feng, Heng Guo, and Jiaheng Wang. Swendsen-wang dynamics for the ferromagnetic ising model with external fields. Information and Computation, 294:105066, 2023. URL: https://doi.org/10.1016/j.ic.2023.105066.
  11. Andreas Galanis, Leslie A Goldberg, and Andrés Herrera-Poyatos. The complexity of approximating the complex-valued ising model on bounded degree graphs. SIAM Journal on Discrete Mathematics, 36(3):2159-2204, 2022. URL: https://doi.org/10.1137/21m1454043.
  12. Reza Gheissari and Alistair Sinclair. Spatial mixing and the random-cluster dynamics on lattices. Random Structures & Algorithms, 64(2):490-534, 2024. URL: https://doi.org/10.1002/rsa.21191.
  13. Heng Guo, Jingcheng Liu, and Pinyan Lu. Zeros of ferromagnetic 2-spin systems. In ACM-SIAM Symposium on Discrete Algorithms, pages 181-192. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.11.
  14. Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on computing, 22(5):1087-1116, 1993. URL: https://doi.org/10.1137/0222066.
  15. Tsung-Dao Lee and Chen-Ning Yang. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Physical Review, 87(3):410, 1952. Google Scholar
  16. Liang Li, Pinyan Lu, and Yitong Yin. Approximate counting via correlation decay in spin systems. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 922-940. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.74.
  17. Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 67-84. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.5.
  18. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. Fisher zeros and correlation decay in the Ising model. Journal of Mathematical Physics, 60(10), 2019. Google Scholar
  19. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. The ising partition function: Zeros and deterministic approximation. Journal of Statistical Physics, 174(2):287-315, 2019. Google Scholar
  20. Ryan L Mann and Michael J Bremner. Approximation algorithms for complex-valued ising models on bounded degree graphs. Quantum, 3:162, 2019. URL: https://doi.org/10.22331/q-2019-07-11-162.
  21. Viresh Patel and Guus Regts. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing, 46(6):1893-1919, 2017. URL: https://doi.org/10.1137/16M1101003.
  22. Viresh Patel, Guus Regts, and Ayla Stam. A near-optimal zero-free disk for the ising model. ArXiv, abs/2311.05574, 2023. URL: https://doi.org/10.48550/arXiv.2311.05574.
  23. Han Peters and Guus Regts. On a conjecture of sokal concerning roots of the independence polynomial. Michigan Mathematical Journal, 68(1):33-55, 2019. Google Scholar
  24. Han Peters and Guus Regts. Location of zeros for the partition function of the Ising model on bounded degree graphs. Journal of the London Mathematical Society, 101(2):765-785, 2020. Google Scholar
  25. Guus Regts. Absence of zeros implies strong spatial mixing. Probability Theory and Related Fields, 186(1):621-641, 2023. Google Scholar
  26. Shuai Shao and Ke Shi. Zero-freeness is all you need: A weitz-type fptas for the entire lee-yang zero-free region. arXiv preprint arXiv:2509.06623, 2025. URL: https://doi.org/10.48550/arXiv.2509.06623.
  27. Shuai Shao and Yuxin Sun. Contraction: A unified perspective of correlation decay and zero-freeness of 2-spin systems. Journal of Statistical Physics, 185:1-25, 2021. Google Scholar
  28. Shuai Shao and Xiaowei Ye. From zero-freeness to strong spatial mixing via a christoffel-darboux type identity. arXiv preprint arXiv:2401.09317, 2024. URL: https://doi.org/10.48550/arXiv.2401.09317.
  29. Alistair Sinclair, Piyush Srivastava, and Marc Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. Journal of Statistical Physics, 155(4):666-686, 2014. Google Scholar
  30. Joachim Von Zur Gathen and Jürgen Gerhard. Modern computer algebra. Cambridge university press, 2003. Google Scholar
  31. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 140-149, 2006. URL: https://doi.org/10.1145/1132516.1132538.
  32. Chen-Ning Yang and Tsung-Dao Lee. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Physical Review, 87(3):404, 1952. Google Scholar
  33. Jinshan Zhang, Heng Liang, and Fengshan Bai. Approximating partition functions of the two-state spin system. Information Processing Letters, 111(14):702-710, 2011. URL: https://doi.org/10.1016/j.ipl.2011.04.012.
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