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Critical correlation functions for the 4-dimensional weakly self-avoiding walk and \(n\)-component \(|\varphi|^4\) model. (English) Zbl 1342.82070
The authors study the critical behavior of the continuous-time weakly self-avoiding walk and of the \(n\)-component \(|\varphi|^4\) model, for all \(n\geq 1\), in the upper critical dimension \(d=4\). Using a rigorous renormalization group method the corresponding critical correlation functions are studied. It is proven that, for the \(|\varphi|^4\) model the critical two-point function has \(|x|^{-2}\) (Gaussian) decay asymptotically, for \(n\geq 1\). Moreover the authors determine the asymptotic decay of the critical correlations of the squares of components of \(\varphi\), including the logarithmic corrections to Gaussian scaling, for \(n\geq 1\). For the continuous-time weakly self-avoiding walk, the authors determine the decay of the critical generating function for the “watermelon” network consisting of \(p\) weakly mutually-and self-avoiding walks, for all \(p\geq 1\), including the logarithmic corrections. For both models, the approach to the critical point is studied and the existence of logarithmic corrections to scaling for certain correlation functions is proved.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
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