Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis

Abstract

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on ${\mathbb{Z}}^d$, in the critical dimension $d=4$, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent 1/4 for the logarithm. The susceptibility has been well understood previously for dimensions $d\ge 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

63 pages, revised version, will appear in Commun. Math. Phys

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Cited by in corpus (37)

• 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
• A renormalisation group method. IV. Stability 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
• Structural stability of a dynamical system near a non-hyperbolic fixed point
• Compressed self-avoiding walks, bridges and polygons
• Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\phi {|}^4$ model
• Self-avoiding walk, spin systems, and renormalization
• Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|\phi {|}^4$ spins
• Mean-field behavior for nearest-neighbor percolation in $d>10$
• A path-integral analysis of interacting Bose gases and loop gases
• Cauchy-Born Rule from Microscopic Models with Non-convex Potentials
• Logarithmic finite-size scaling of the self-avoiding walk at four dimensions
• Weakly self-avoiding walk on a high-dimensional torus
• Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps
• Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
• Finite Range Decomposition for Gaussian Measures with Improved Regularity
• Spectral gap critical exponent for Glauber dynamics of hierarchical spin models
• The geometry of random walk isomorphism theorems
• Four-dimensional weakly self-avoiding walk with contact self-attraction
• Three-dimensional tricritical spins and polymers
• On self-avoiding polygons and walks: counting, joining and closing
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• On the existence of critical exponents for self-avoiding walks
• Renormalisation group analysis of 4D spin models and self-avoiding walk
• On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian
• Percolation transition for random forests in $d\ge 3$
• Spin systems with hyperbolic symmetry: a survey
• Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed
• An upper bound on the number of self-avoiding polygons via joining