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Spectral gap critical exponent for Glauber dynamics of hierarchical spin models

Presented by: 
Roland Bauerschmidt
Friday 7th September 2018 - 09:00 to 10:00
INI Seminar Room 1
We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems at and near a critical point. In our approach, we derive a spectral gap inequality, or more generally a Brascamp--Lieb inequality, for the measure recursively in terms of spectral gap or Brascamp--Lieb inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|\varphi|^4$ model at the critical point and its approach from the high temperature side, and the $2$-dimensional Sine--Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz--Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the $|\varphi|^4$ model), the scaling limit of these models in equilibrium. Co-author: Thierry Bodineau
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons