09-6 David Ruelle
Characterization of Lee-Yang polynomials (42K, plain TeX) Jan 7, 09
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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=\ldots=z_n=z\$ in certain multiaffine polynomials \$\Psi(z_1,\ldots,z_n)\$ which we call Lee-Yang polynomials (they do not vanish when \$|z_1|,\ldots,|z_n|<1\$ or \$|z_1|,\ldots,|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,\ldots,z_n)\ne0\$ when \$|z_1|,\ldots,|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 Lee and Yang.

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