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Scaling limits and critical behaviour of the 4-dimensional \(n\)-component \(|\varphi|^4\) spin model. (English) Zbl 1308.82026
Summary: We consider the \(n\)-component \(|\varphi|^4\) spin model on \(\mathbb Z^4\), for all \(n\geq 1\), with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\); double logarithmic scaling for \(n=4\); and is bounded when \(n>4\). In addition, for the model defined on the \(4\)-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi|^4\) model.

MSC:
82B28 Renormalization group methods in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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Ptsoft
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