A renormalisation group method. IV. Stability analysis

Abstract

This paper is the fourth in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The third paper in the series presents a perturbative analysis of a supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on ${\mathbb{Z}}^d$. We now present an analysis of the relevant interaction functional of the supersymmetric field theory, which permits a nonperturbative analysis to be carried out in the critical dimension $d=4$. The results in this paper include: proof of stability of the interaction, estimates which enable control of Gaussian expectations involving both boson and fermion fields, estimates which bound the errors in the perturbative analysis, and a crucial contraction estimate to handle irrelevant directions in the flow of the renormalisation group. These results are essential for the analysis of the general renormalisation group step in the fifth paper in the series.

62 pages

References in corpus (17)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• A Complete Renormalization Group Trajectory Between Two Fixed Points
• Sine-Gordon Revisited
• A renormalisation group method. II. Approximation by local polynomials
• Functional integral representations for self-avoiding walk
• A renormalisation group method. III. Perturbative analysis
• A simple method for finite range decomposition of quadratic forms and Gaussian fields
• The Renormalization Group According to Balaban - I. Small fields
• A renormalisation group method. I. Gaussian integration and normed algebras
• A renormalisation group method. V. A single renormalisation group step
• Structural stability of a dynamical system near a non-hyperbolic fixed point
• Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas
• 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
• Infinite volume limit for the dipole gas

Cited by in corpus (23)

• A variational method for ${\Phi }_3^4$
• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• Introduction to a renormalisation group method
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• A renormalisation group method. II. Approximation by local polynomials
• Critical exponents for long-range O(n) models below the upper critical dimension
• A renormalisation group method. III. Perturbative analysis
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension
• A renormalisation group method. I. Gaussian integration and normed algebras
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|\phi {|}^4$ spins
• Spectral gap critical exponent for Glauber dynamics of hierarchical spin models
• Four-dimensional weakly self-avoiding walk with contact self-attraction
• On self-avoiding polygons and walks: counting, joining and closing
• Three-dimensional tricritical spins and polymers
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• Renormalisation group analysis of 4D spin models and self-avoiding walk
• Percolation transition for random forests in $d\ge 3$
• Scaling limit and strict convexity of free energy for gradient models with non-convex potential
• A renormalization group method by harmonic extensions and the classical dipole gas
• Decay of covariance for gradient models with non-convex potential