Skip to main content
SpringerLink
Log in

The Ising Partition Function: Zeros and Deterministic Approximation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. If a combinatorial counting problem, such as computing a partition function in a statistical physics model, is #P-hard, then it can be solved in polynomial time only if all counting problems belonging to a very rich class can be solved in polynomial time. Hence #P-hardness is widely regarded as compelling evidence of the intractibility of efficient exact computation. For a more detailed account of this phenomenon in the context of partition functions, see, e.g., [47, Appendix A].

  2. An FPTAS takes as input an n-vertex (hyper)graph G, model parameters \(\beta ,\lambda \), and an accuracy parameter \(\varepsilon \in (0,1)\) and outputs a value that approximates \(Z_G^\beta (\lambda )\) within a factor \(1\pm \varepsilon \) [see also Eq. (3)]. The running time of the algorithm is polynomial in n and \(1/\varepsilon \).

  3. A quasi-polynomial time algorithm is one which runs in time \(\exp \{O((\log n)^c)\}\) for some constant \(c>1\).

References

  1. Anari, N., Gharan, S.O.: The Kadison–Singer problem for strongly Rayleigh measures and applications to Asymmetric TSP. In: Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS) (2015)

  2. Anari, N., Gharan, S.O.: A generalization of permanent inequalities and applications in counting and optimization. In: Proceedings of the 49th ACM Symposium on Theory and Computation (STOC), pp. 384–396. arXiv:1702.02937 (2017)

  3. Asano, T.: Lee–Yang theorem and the Griffiths inequality for the anisotropic Heisenberg ferromagnet. Phys. Rev. Let. 24(25), 1409–1411 (1970). https://doi.org/10.1103/PhysRevLett.24.1409

    Article  ADS  MathSciNet  Google Scholar 

  4. Barata, J.C.A., Goldbaum, P.S.: On the distribution and gap structure of Lee–Yang zeros for the Ising model: periodic and aperiodic couplings. J. Stat. Phys. 103(5–6), 857–891 (2001). https://doi.org/10.1023/A:1010332500031

    Article  MathSciNet  MATH  Google Scholar 

  5. Barata, J.C.A., Marchetti, D.H.U.: Griffiths’ singularities in diluted Ising models on the Cayley tree. J. Stat. Phys. 88(1–2), 231–268 (1997). https://doi.org/10.1007/BF02508471

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Barvinok, A.: Computing the partition function for cliques in a graph. Theory Comput. 11(13), 339–355 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barvinok, A.: Computing the permanent of (some) complex matrices. Found. Comput. Math. 16(2), 329–342 (2015). https://doi.org/10.1007/s10208-014-9243-7

    Article  MathSciNet  MATH  Google Scholar 

  8. Barvinok, A.: Combinatorics and Complexity of Partition Functions. Springer, New York (2016)

    Book  MATH  Google Scholar 

  9. Barvinok, A., Soberón, P.: Computing the partition function for graph homomorphisms with multiplicities. J. Combin. Theory Ser. A 137, 1–26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barvinok, A., Soberón, P.: Computing the partition function for graph homomorphisms. Combinatorica 37(4), 633–650 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berger, N., Kenyon, C., Mossel, E., Peres, Y.: Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131(3), 311–340 (2005). https://doi.org/10.1007/s00440-004-0369-4

    Article  MathSciNet  MATH  Google Scholar 

  12. Borcea, J., Brändén, P.: The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving stability. Invent. Math. 177(3), 541–569 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Borcea, J., Brändén, P.: The Lee–Yang and Pólya–Schur programs. II. Theory of stable polynomials and applications. Commun. Pure Appl. Math. 62(12), 1595–1631 (2009)

    Article  MATH  Google Scholar 

  14. Borcea, J., Brändén, P., Liggett, T.: Negative dependence and the geometry of polynomials. J. AMS 22(2), 521–567 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Borgs, C., Chayes, J., Kahn, J., Lovász, L.: Left and right convergence of graphs with bounded degree. Random Struct. Algorithms 42(1), 1–28 (2013). https://doi.org/10.1002/rsa.20414

    Article  MathSciNet  MATH  Google Scholar 

  16. Cai, J.Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: a dichotomy theorem. In: Proceedings of the ICALP, Lecture Notes in Computer Science, vol. 6198, pp. 275–286. Springer. http://www.springerlink.com/content/46275700132p5250/abstract/ (2010)

  17. Csikvári, P., Frenkel, P.E.: Benjamini–Schramm continuity of root moments of graph polynomials. Eur. J. Combin. 52, 302–320 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Efthymiou, C., Hayes, T.P., Štefankovic, D., Vigoda, E., Yin, Y.: Convergence of MCMC and loopy BP in the tree uniqueness region for the hard-core model. In: Proceedings of the 57th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 704–713 (2016)

  19. Galanis, A., Goldberg, L.A.: The complexity of approximately counting in 2-spin systems on \(k\)-uniform bounded-degree hypergraphs. Inf. Comput. 251, 36–66 (2016). https://doi.org/10.1016/j.ic.2016.07.003. http://www.sciencedirect.com/science/article/pii/S0890540116300426

  20. Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. Random Struct. Algorithms 45(1), 78–110 (2014). https://doi.org/10.1002/rsa.20479

    Article  MathSciNet  MATH  Google Scholar 

  21. Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Inf. Comput. 206(7), 908–929 (2008). https://doi.org/10.1016/j.ic.2008.04.003. http://www.sciencedirect.com/science/article/pii/S089054010800031X

  22. Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. SIAM J. Comput. 39(7), 3336–3402 (2010). https://doi.org/10.1137/090757496

    Article  MathSciNet  MATH  Google Scholar 

  23. Goldberg, L.A., Jerrum, M., Paterson, M.: The computational complexity of two-state spin systems. Random Struct. Algorithms 23(2), 133–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guo, H., Jerrum, M.: Random cluster dynamics for the Ising model is rapidly mixing. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1818–1827 (2017)

  25. Guo, H., Lu, P.: Uniqueness, spatial mixing, and approximation for ferromagnetic 2-spin systems. In: Proceedings of the APPROX/RANDOM (2016)

  26. Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258 (1925). https://doi.org/10.1007/BF02980577

    Article  ADS  Google Scholar 

  27. Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989). https://doi.org/10.1137/0218077

    Article  MathSciNet  MATH  Google Scholar 

  28. Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22(5), 1087–1116 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jerrum, M., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87(3), 410–419 (1952). https://doi.org/10.1103/PhysRev.87.410

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 67–84 (2013)

  32. Long, Y., Nachmias, A., Ning, W., Peres, Y.: A power law of order 1/4 for critical mean-field Swendsen–Wang dynamics. AMS MEMO/232/1092 (2014)

  33. Lu, P., Yang, K., Zhang, C.: FPTAS for hardcore and Ising models on hypergraphs. In: Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS), pp. 51:1–51:14 (2016)

  34. Lubetzky, E., Sly, A.: Critical Ising on the square lattice mixes in polynomial time. Commun. Math. Phys. 313(3), 815–836 (2012). https://doi.org/10.1007/s00220-012-1460-9

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Luby, M., Vigoda, E.: Approximately counting up to four. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 682–687. https://doi.org/10.1145/258533.258663 (1997)

  36. Marcus, A., Spielman, D., Srivastava, N.: Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. Math. (2015) https://doi.org/10.4007/annals.2015.182.1.7. http://annals.math.princeton.edu/2015/182-1/p07

  37. Marcus, A.W., Spielman, D.A., Srivastava, N.: Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem. Ann. Math. 182, 327–350 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Commun. Math. Phys. 161(3), 447–486 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region: II. The general case. Commun. Math. Phys. 161, 487–514 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Martinelli, F., Sinclair, A., Weitz, D.: Glauber dynamics on trees: Boundary conditions and mixing time. Commun. Math. Phys. 250(2), 301–334 (2004). https://doi.org/10.1007/s00220-004-1147-y

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Mossel, E., Sly, A.: Exact thresholds for Ising–Gibbs samplers on general graphs. Ann. Probab. 41(1), 294–328 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Patel, V., Regts, G.: Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM J. Comput. 46(6), 1893–1919 (2017). arXiv:1607.01167

    Article  MathSciNet  MATH  Google Scholar 

  43. Randall, D., Wilson, D.: Sampling spin configurations of an Ising system. In: Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 959–960 (1999)

  44. Ruelle, D.: Characterization of Lee-Yang polynomials. Ann. Math. 171(1), 589–603 (2010). https://doi.org/10.4007/annals.2010.171.589. http://annals.math.princeton.edu/2010/171-1/p16

  45. Scott, A., Sokal, A.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118(5–6), 1151–1261 (2004)

    ADS  MATH  Google Scholar 

  46. Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sinclair, A., Srivastava, P.: Lee–Yang theorems and the complexity of computing averages. Commun. Math. Phys. 329(3), 827–858 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Sinclair, A., Srivastava, P., Thurley, M.: Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. J. Stat. Phys. 155(4), 666–686 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Sinclair, A., Srivastava, P., Štefankovič, D., Yin, Y.: Spatial mixing and the connective constant: optimal bounds. Probab. Theory Relat. Fields 168, 153–197 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sly, A., Sun, N.: Counting in two-spin models on \(d\)-regular graphs. Ann. Probab. 42(6), 2383–2416 (2014). https://doi.org/10.1214/13-AOP888. http://projecteuclid.org/euclid.aop/1412083628

  51. Song, R., Yin, Y., Zhao, J.: Counting hypergraph matchings up to uniqueness threshold. In: Proceedings of the APPROX/RANDOM, pp. 46:1–46:29 (2016)

  52. Stanley, R., Fomin, S.: Enumerative Combinatorics. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  53. Straszak, D., Vishnoi, N.K.: Real stable polynomials and matroids: optimization and counting. In: Proceedings of the 49th ACM Symposium on Theory of Computing (STOC), pp. 370–383. arXiv:1611.04548 (2017)

  54. Suzuki, M., Fisher, M.E.: Zeros of the partition function for the Heisenberg, Ferroelectric, and general Ising models. J. Math. Phys. 12(2), 235–246 (1971). https://doi.org/10.1063/1.1665583. http://scitation.aip.org/content/aip/journal/jmp/12/2/10.1063/1.1665583;jsessionid=tsaFQsrLOe4npdBWQR-8iADE.x-aip-live-06

  55. Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of the 38th ACM Symposium on Theory of Computing (STOC), pp. 140–149 (2006)

  56. Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87(3), 404–409 (1952). https://doi.org/10.1103/PhysRev.87.404

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Zhang, J., Liang, H., Bai, F.: Approximating partition functions of the two-state spin system. Inf. Process. Lett. 111(14), 702–710 (2011). https://doi.org/10.1016/j.ipl.2011.04.012

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Alexander Barvinok, Guus Regts and anonymous reviewers for helpful comments.

Author information

Authors and Affiliations

  1. Computer Science Division, UC Berkeley, Berkeley, USA

    Jingcheng Liu & Alistair Sinclair

  2. Tata Institute of Fundamental Research, Mumbai, India

    Piyush Srivastava

Authors
  1. Jingcheng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alistair Sinclair
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Piyush Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alistair Sinclair.

Additional information

JL and AS are supported in part by US NSF Grants CCF-1420934 and CCF-1815328. PS is supported by a Ramanujan Fellowship of the Indian Department of Science and Technology. Some of this work was done at the Simons Institute for the Theory of Computing at UC Berkeley.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Sinclair, A. & Srivastava, P. The Ising Partition Function: Zeros and Deterministic Approximation. J Stat Phys 174, 287–315 (2019). https://doi.org/10.1007/s10955-018-2199-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-2199-2

Keywords

Search

Navigation