Characterization of Lee-Yang polynomials

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 $z_1=\dots=z_n=z$ 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 $\Phi(z_1,\dots,z_n)\ne0$ 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 Lee and Yang.