Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics

Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics

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Mikhail Lyubich, SUNY at Stony Brook
Fine Hall 401

In a classical work of 1950's, Lee and Yang proved that zeros of the partition functions of the Ising models on graphs 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 a special "Diamond Hierarchical Lattice." In this case, it can be described in terms of the dynamics of an explicit rational map in two variables. We prove partial hyperbolicity of this map on an invariant cylinder, and derive from it that the Lee-Yang zeros are organized asymptotically in a transverse measure for the central foliation. From the global complex point of view, the zero distributions get interpreted as slices of the Green (1,1)-current on the projective space. It is a joint work with Pavel Bleher and Roland Roeder.