Mathematics – Dynamical Systems
Scientific paper
2010-09-23
Mathematics
Dynamical Systems
99 pages
Scientific paper
In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\RR$ in two variables (the renormalization transformation). We prove that $\RR$ is partially hyperbolic on an invariant cylinder $\CC$. The Lee-Yang zeros are organized in a transverse measure for the central-stable foliation of $\RR|\, \CC$. Their distribution is absolutely continuous. Its density is $C^\infty$ (and non-vanishing) below the critical temperature. Above the critical temperature, it is $C^\infty$ on a open dense subset, but it vanishes on the complementary Cantor set of positive measure. This seems to be the first occasion of a complete rigorous description of the Lee-Yang distributions beyond 1D models.
Bleher Pavel
Lyubich Mikhail
Roeder Roland
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