Abstract
The IsingKac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius γ^{− 1} for « 1 around its base point. We study the Glauber dynamics for this model on a discrete twodimensional torus Z^{2}/(2N + 1)Z^{2} for a system size « 1 γ^{1} and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarsegrained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation. This equation is the dynamic version of the Ф_{4} ^{2} quantum field theory, which is formally given by a reactiondiffusion equation driven by an additive spacetime white noise. It is wellknown that in two spatial dimensions such equations are distribution valued and a Wick renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.
Original language  English 

Pages (fromto)  717812 
Number of pages  96 
Journal  Communications on Pure and Applied Mathematics 
Volume  70 
Issue number  4 
Early online date  20 Aug 2016 
DOIs  
Publication status  Published  21 Feb 2017 
ASJC Scopus subject areas
 Mathematics(all)
 Applied Mathematics
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Hendrik Weber
 Department of Mathematical Sciences  Professor of Probability
 Probability Laboratory at Bath
Person: Research & Teaching