Showing papers by "David Ruelle published in 2010"
TL;DR: The Lee-Yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle jzj D 1 as mentioned in this paper, which are obtained by taking z1 D D zn D z in certain multiaffine polynomorphs.
Abstract: The Lee-Yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle jzj D 1. These polynomials are obtained by taking z1 D D zn D z in certain multiaffine polynomials ‰.z1; : : : ; zn/ which we call Lee-Yang polynomials (they do not vanish when jz1j; : : : ; jznj 1). We characterize the Lee-Yang polynomials ‰ in nC1 variables in terms of polynomials ˆ in n variables (those such that ˆ.z1; : : : ; zn/¤ 0 when jz1j; : : : ; jznj 1 (including jzi j D 1 in a sense to be made precise later). Our current understanding of Lee-Yang polynomials is based on the concept of Asano contraction [1]. We shall define an inner radius associated with a multiaffine polynomial ˆ, and see that it behaves supermultiplicatively with respect to Asano contraction (Proposition 2). Using the properties of the inner radius, we shall characterize the ‰ 2 LYnC1 (nC 1 variables) in terms of polynomials ˆ in n variables such that ˆ.z1; : : : ; zn/¤ 0 when jz1j; : : : ; jznj < 1 (Theorem 3). This characterization will give us a good understanding of LYnC1 (Proposition 5), and allow us to exhibit elements of LYnC1 outside of the (pair interaction) class originally considered by Lee and Yang (see in particular Example 7(d)). The original
41 citations
TL;DR: In this article, it has been shown that the susceptibility function of the power series has a radius of convergence with respect to a given parameter if ρ ≥ 1/2 and ρ = ρ(1/2).
Abstract: Let $\rho$ be an SRB (or "physical"), measure for the discrete time evolution given by a map $f$, and let $\rho(A)$ denote the expectation value of a smooth function $A$. If $f$ depends on a parameter, the derivative $\delta\rho(A)$ of $\rho(A)$ with respect to the parameter is formally given by the value of the so-called susceptibility function $\Psi(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $\Psi(z)$ has a radius of convergence $r(\Psi)>1$, and that $\delta\rho(A)=\Psi(1)$, but it is known that $r(\Psi) 1$ if $d>1/2$. In particular, if $d>1/2$ we may hope that $\Psi(1)$ makes sense, and the derivative $\delta\rho(A)=\Psi(1)$ has thus a chance to be defined
5 citations