Inventiones mathematicae

, 177:541 | Cite as

The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability

  • Julius Borcea
  • Petter BrändénEmail author


In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Pólya-Schur on univariate polynomials with such properties.

Mathematics Subject Classification (2000)

47B38 05A15 05C70 30C15 32A60 46E22 82B20 82B26 


  1. 1.
    Asano, T.: Theorems on the partition functions of the Heisenberg ferromagnets. J. Phys. Soc. Jpn. 29, 350–359 (1970) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Atiyah, M.F., Bott, R., Gårding, L.: Lacunas for hyperbolic differential operators with constant coefficients I. Acta Math. 124, 109–189 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beauzamy, B.: On complex Lee and Yang polynomials. Commun. Math. Phys. 182, 177–184 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Biskup, M., Borgs, C., Chayes, J.T., Kleinwaks, L.J., Kotecky, R.: Partition function zeros at first-order phase transitions: A general analysis. Commun. Math. Phys. 251, 79–131 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Biskup, M., Borgs, C., Chayes, J.T., Kotecky, R.: Partition function zeros at first-order phase transitions: Pirogov-Sinai theory. J. Stat. Phys. 116, 97–155 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borcea, J., Brändén, P.: Applications of stable polynomials to mixed determinants: Johnson’s conjectures, unimodality, and symmetrized Fischer products. Duke Math. J. 143, 205–223 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Borcea, J., Brändén, P.: The Lee-Yang and Pólya-Schur programs. II. Theory of stable polynomials and applications. arXiv:0809.3087
  8. 8.
    Borcea, J., Brändén, P.: Pólya-Schur master theorems for circular domains and their boundaries. Ann. Math. (to appear). arXiv:math/0607416
  9. 9.
    Borcea, J., Brändén, P.: Multivariate Pólya-Schur classification problems in the Weyl algebra. arXiv:math/0606360
  10. 10.
    Borcea, J., Brändén, P., Liggett, T.M.: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22, 521–567 (2009). arXiv:0707.2340 CrossRefGoogle Scholar
  11. 11.
    Borcea, J., Brändén, P., Csordas, G., Vinnikov, V.: Pólya-Schur-Lax problems: hyperbolicity and stability preservers, Workshop Report, American Institute of Mathematics, Palo Alto, CA, May–June 2007.
  12. 12.
    Brändén, P.: Polynomials with the half-plane property and matroid theory. Adv. Math. 216, 302–320 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Choe, Y., Oxley, J., Sokal, A.D., Wagner, D.G.: Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. 32, 88–187 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Craven, T., Csordas, G.: Multiplier sequences for fields. Ill. J. Math. 21, 801–817 (1977) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Craven, T., Csordas, G.: Composition theorems, multiplier sequences and complex zero decreasing sequences. In: Barsegian, G., Laine, I., Yang, C.C. (eds.) Value Distribution Theory and Its Related Topics, pp. 131–166. Kluwer, Dordrecht (2004) CrossRefGoogle Scholar
  16. 16.
    Craven, T., Csordas, G., Smith, W.: The zeros of derivatives of entire functions and the Pólya-Wiman conjecture. Ann. Math. (2) 125, 405–431 (1987) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Csordas, G.: Linear operators and the distribution of zeros of entire functions. Complex Var. Elliptic Equ. 51, 625–632 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Edrei, A.: Power series having partial sums with zeros in a half-plane. Proc. Am. Math. Soc. 9, 320–324 (1958) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Fisk, S.: Polynomials, roots, and interlacing. Versions 1–2., xx+700 pp
  20. 20.
    Gårding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Grace, J.H.: The zeros of a polynomial. Proc. Camb. Philos. Soc. 11, 352–357 (1902) Google Scholar
  22. 22.
    Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Commun. Math. Phys. 25, 190–232 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hinkkanen, A.: Schur products of certain polynomials. In: Dodziuk, J., Keenin, L. (eds.) Lipa’s Legacy: Proceedings of the Bers Colloquium. Contemp. Math., vol. 211, pp. 285–295. Am. Math. Soc., Providence (1997) Google Scholar
  24. 24.
    Hörmander, L.: Notions of Convexity. Progr. Math., vol. 127. Birkhäuser, Boston (1994) Google Scholar
  25. 25.
    Iserles, A., Nørsett, S.P., Saff, E.B.: On transformations and zeros of polynomials. Rocky Mt. J. Math. 21, 331–357 (1991) zbMATHCrossRefGoogle Scholar
  26. 26.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163(2), 1019–1056 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Laguerre, E.: Fonctions du genre zéro et du genre un. C. R. Acad. Sci. Paris 95, 828–831 (1882) Google Scholar
  28. 28.
    Lax, P.D.: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 6, 175–194 (1958) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87, 410–419 (1952) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Levin, B.Ja.: Distribution of Zeros of Entire Functions. Transl. Math. Monogr., vol. 5. Am. Math. Soc., Providence (1980) Google Scholar
  31. 31.
    Lieb, E.H., Sokal, A.D.: A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Commun. Math. Phys. 80, 153–179 (1981) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Liggett, T.M.: Distributional limits for the symmetric exclusion process. Stoch. Process. Appl. 119, 1–15 (2009). arXiv:0710.3606 zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Math. Surveys, vol. 3. Am. Math. Soc., New York (1949) zbMATHGoogle Scholar
  34. 34.
    Newman, C.M.: Zeros of the partition function for generalized Ising systems. Commun. Pure Appl. Math. 27, 143–159 (1974) CrossRefGoogle Scholar
  35. 35.
    Newman, C.M.: Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Commun. Math. Phys. 41, 1–9 (1975) CrossRefGoogle Scholar
  36. 36.
    Pólya, G.: Bemerkung über die Integraldarstellung der Riemannsche ξ-Funktion. Acta Math. 48, 305–317 (1926) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Pólya, G.: Collected Papers, vol. II: Location of Zeros. Mathematicians of our Time, vol. 8. MIT Press, Cambridge (1974). ed. R.P. Boas Google Scholar
  38. 38.
    Pólya, G., Schur, I.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914) zbMATHGoogle Scholar
  39. 39.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. II. Springer, Berlin (1976) Google Scholar
  40. 40.
    Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. London Math. Soc. Monogr. (N.S.), vol. 26. Oxford Univ. Press, New York (2002) zbMATHGoogle Scholar
  41. 41.
    Ruelle, D.: Extension of the Lee–Yang circle theorem. Phys. Rev. Lett. 26, 303–304 (1971) CrossRefMathSciNetGoogle Scholar
  42. 42.
    Ruelle, D.: Is our mathematics natural? The case of equilibrium statistical mechanics. Bull. Am. Math. Soc. (N.S.) 19, 259–268 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Ruelle, D.: Zeros of graph-counting polynomials. Commun. Math. Phys. 200, 43–56 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. World Scientific, River Edge (1999). Reprint of the 1989 edition Google Scholar
  45. 45.
    Ruelle, D.: Grace-like polynomials. In: Foundations of Computational Mathematics, Hong Kong, 2000, pp. 405–421. World Scientific, River Edge (2002) Google Scholar
  46. 46.
    Schur, I.: Zwei Sätze über algebraische Gleichungen mit lauter reellen Wurzeln. J. Reine Angew. Math. 144, 75–88 (1923) Google Scholar
  47. 47.
    Scott, A.D., Sokal, A.D.: The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118, 1151–1261 (2005). arXiv:cond-mat/0309352 zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Sokal, A.D.: Chromatic roots are dense in the whole complex plane. Comb. Probab. Comput. 13, 221–261 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Webb, B.S. (ed.) Surveys in Combinatorics, 2005. Cambridge Univ. Press, Cambridge (2005). arXiv:math.CO/0503607 Google Scholar
  50. 50.
    Szász, O.: On sequences of polynomials and the distribution of their zeros. Bull. Am. Math. Soc. 49, 377–383 (1943) zbMATHCrossRefGoogle Scholar
  51. 51.
    Szegö, G.: Bemerkungen zu einem Satz von J.H. Grace über die Wurzeln algebraischer Gleichungen. Math. Z. 13, 28–55 (1922) CrossRefMathSciNetGoogle Scholar
  52. 52.
    Wagner, D.G.: Weighted enumeration of spanning subgraphs with degree constraints. arXiv:0803.1659
  53. 53.
    Walsh, J.L.: On the location of the roots of certain types of polynomials. Trans. Am. Math. Soc. 24, 163–180 (1922) CrossRefGoogle Scholar
  54. 54.
    Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87, 404–409 (1952) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

Personalised recommendations