Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model

Abstract

We consider the $n$-component $|\phi {|}^4$ spin model on ${\mathbb{Z}}^4$, for all $n\ge 1$, 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 $|\phi {|}^4$ model.

55 pages

References in corpus (16)

• Critical behaviour of the Ising model on the 4-dimensional lattice
• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• A Complete Renormalization Group Trajectory Between Two Fixed Points
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• A renormalisation group method. II. Approximation by local polynomials
• A renormalisation group method. III. Perturbative analysis
• A simple method for finite range decomposition of quadratic forms and Gaussian fields
• A renormalisation group method. IV. Stability analysis
• A renormalisation group method. I. Gaussian integration and normed algebras
• Structural stability of a dynamical system near a non-hyperbolic fixed point
• Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions
• Application of the lace expansion to the ${\phi }^4$ model
• The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
• Critical exponents of the two dimensional Coulomb gas at the Berezinskii-Kosterlitz-Thouless transition
• Strict Convexity of the Surface Tension for Non-convex Potentials

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• Critical exponents for long-range O(n) models below the upper critical dimension
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• A renormalisation group method. I. Gaussian integration and normed algebras
• Scaling at quantum phase transitions above the upper critical dimension
• A renormalisation group method. V. A single renormalisation group step
• Structural stability of a dynamical system near a non-hyperbolic fixed point
• Application of the lace expansion to the ${\phi }^4$ model
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