2010/09/23 by Pavel Bleher, Bleher, Pavel, Mikhail Lyubich +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1009.4691
openalex publication_date 2010/09/23 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In a classical work of the 1950's, Lee and Yang proved that the zeros of the\npartition functions of a ferromagnetic Ising models always lie on the unit\ncircle. Distribution of these zeros is physically important as it controls\nphase transitions in the model. We study this distribution for the\nMigdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be\ndescribed in terms of the dynamics of an explicit rational function RR in\ntwo variables (the renormalization transformation). We prove that RR is\npartially hyperbolic on an invariant cylinder CC. The Lee-Yang zeros are\norganized in a transverse measure for the central-stable foliation of RR| ,\n CC. Their distribution is absolutely continuous. Its density is C^\∞\n(and non-vanishing) below the critical temperature. Above the critical\ntemperature, it is C^\∞ on a open dense subset, but it vanishes on the\ncomplementary Cantor set of positive measure. This seems to be the first\noccasion of a complete rigorous description of the Lee-Yang distributions\nbeyond 1D models.\n