Characterization of Lee-Yang polynomials


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.


David Ruelle

Department of Mathematics, Rutgers University, 110 Frelinghuson Rd, Piscataway, NJ 08544-8019, United States and Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, 91440 Bures Sur Yvette, France