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Mathematics Colloquium

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Date:  Wednesday, April 07, 2010

Title:  Lee-Yang zeros and Enumerative Dynamics

Abstract:  In a classical work of 1952, Lee and Yang proved that zeros of certain polynomials (the ``partition functions" of Ising models) always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. However, it is not an easy task to describe this distribution, so little is known about it beyond the one-dimensional model. Jointly with Pavel Bleher and Roland Roeder, we have studied distribution of the Lee-Yang zeros for a special ``Diamond Hierarchical Lattice" which can be viewed as an approximation to the regular 2D lattice. In this case, it can be described in terms of dynamics of an explicit rational map in two variables acting on an invariant cylinder. We prove that the Lee-Yang zeros of this model are organized in a transverse measure for a dynamical foliation on the cylinder. Moreover, transverse regularity of this foliation reflects phase transitions in the model. From the global complex point of view, these measures get interpreted as slices of the Green (1,1)-current on the projective space. The proofs are based on a mixture of methods from real and complex dynamics that exploit ideas of Kobayashi hyperbolicity and geometry of algebraic curves.


Speaker:  Mikhail Lyubich
Institution:  SUNY Stonybrook

 

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