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Characterization of Lee-Yang polynomials. (English) Zbl 1251.30005

Author’s abstract: “The Lee-Yang circle theorem describes complex polynomials of degree \(n\) in \(z\) with all their zeros on the unit circle \(|z|=1\). These polynomials are obtained by taking \(z_1=\cdots=z_n=z\) in certain multiaffine polynomials \(\Psi(z_1,\dots,z_n)\) which we call Lee-Yang polynomials (they do not vanish when \(|z_1|,\dots,|z_n| < 1\) or \(|z_1|,\dots,|z_n| > 1\)). We characterize the Lee-Yang polynomials \(\Psi\) in \(n+1\) variables in terms of polynomials \(\Phi\) in \(n\) variables (those such that \(\Phi(z_1,\dots,z_n)\not= 0\) when \(|z_1|,\dots,|z_n| < 1\)). This characterization gives us a good understanding of Lee-Yang polynomials and allows us to exhibit some new examples. In the physical situation where the \(\Psi\) are temperature dependent partition functions, we find that those \(\Psi\) which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by T. D. Lee and C. N. Yang [Phys. Rev., II. Ser. 87, 410–419 (1952; Zbl 0048.43401)].”
See also the author’s paper [“Extension of the Lee-Yang circle theorem”, Phys. Rev. Lett. 26, 303–304 (1971; doi:10.1103/PhysRevLett.26.870)].

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics

Citations:

Zbl 0048.43401
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References:

[1] T. Asano, ”Theorems on the partition functions of the Heisenberg ferromagnets.,” J. Phys. Soc. Japan, vol. 29, pp. 350-359, 1970. · Zbl 1230.82052 · doi:10.1143/JPSJ.29.350
[2] B. Beauzamy, ”On complex Lee and Yang polynomials,” Comm. Math. Phys., vol. 182, iss. 1, pp. 177-184, 1996. · Zbl 0877.12002 · doi:10.1007/BF02506389
[3] J. Borcea and P. Brändén, ”The Lee-Yang and Polya-Schur programs. I. Linear operators preserving stability,” Invent. Math., vol. 177, pp. 541-569, 2009. · Zbl 1175.47032 · doi:10.1007/s00222-009-0189-3
[4] J. Borcea and P. Brändén, ”The Lee-Yang and Polya-Schur programs. II. Theory of stable polynomials and applications,” Comm. Pure Appl. Math., vol. 62, pp. 1595-1631, 2009. · Zbl 1177.47041 · doi:10.1002/cpa.20295
[5] T. D. Lee and C. N. Yang, ”Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model,” Phys. Rev., vol. 87, pp. 410-419, 1952. · Zbl 0048.43401 · doi:10.1103/PhysRev.87.410
[6] G. Pólya and G. SzegHo, Problems and Theorems in Analysis. II, New York: Springer-Verlag, 1998.
[7] D. Ruelle, ”Extension of the Lee-Yang circle theorem,” Phys. Rev. Lett., vol. 26, pp. 303-304, 1971. · doi:10.1103/PhysRevLett.26.303
[8] D. Ruelle, ”Grace-like polynomials,” in Foundations of Computational Mathematics. Proc. of Smalefest 2000, River Edge, NJ: World Sci. Publ., 2002, p. viii. · Zbl 1173.30304
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