Application of the lace expansion to the ${\phi }^4$ model

Abstract

Using the Griffiths-Simon construction of the ${\phi }^4$ model and the lace expansion for the Ising model, we prove that, if the strength $\lambda \ge 0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, then the critical ${\phi }^4$ two-point function $⟨{\phi }_o{\phi }_x{⟩}_{{\mu }_c}$ is asymptotically $|x{|}^{2-d}$ times a model-dependent constant, and the critical point is estimated as ${\mu }_c=\hat{\mathcal{J}}-\frac{\lambda }{2}⟨{\phi }_o^2{⟩}_{{\mu }_c}+O({\lambda }^2)$, where $\hat{\mathcal{J}}$ is the massless point for the Gaussian model.

32 pages, 5 figures

References in corpus (2)

• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

Cited by in corpus (9)

• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Critical exponents for long-range O(n) models below the upper critical dimension
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• Critical two-point function for long-range models with power-law couplings: The marginal case for $d\ge d_c$
• Gaussian deconvolution and the lace expansion
• The continuous-time lace expansion
• Kotani's Theorem for the Fourier Transform
• Lace expansion for dummies
• Gaussian deconvolution on ${\mathbb{R}}^d$ with application to self-repellent Brownian motion