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Application of the Lace Expansion to the \({\varphi^4}\) Model

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Abstract

Using the Griffiths–Simon construction of the \({\varphi^4}\) model and the lace expansion for the Ising model, we prove that, if the strength \({\lambda\ge0}\) of nonlinearity is sufficiently small for a large class of short-range models in dimensions d > 4, then the critical \({\varphi^4}\) two-point function \({\langle{\varphi_o\varphi_x \rangle}_{\mu_c}}\) is asymptotically \({|x|^{2-d}}\) times a model-dependent constant, and the critical point is estimated as \({\mu_c = \hat{\fancyscript{J}} -\frac{\lambda}{2} \langle {\varphi_o^2}\rangle_{\mu_c} + O (\lambda^2)}\), where \({\hat{\fancyscript{J}}}\) is the massless point for the Gaussian model.

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  1. Department of Mathematics, Hokkaido University, Sapporo, Japan

    Akira Sakai

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  1. Akira Sakai
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Correspondence to Akira Sakai.

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Communicated by H. Spohn

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Sakai, A. Application of the Lace Expansion to the \({\varphi^4}\) Model. Commun. Math. Phys. 336, 619–648 (2015). https://doi.org/10.1007/s00220-014-2256-x

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