A renormalisation group method. I. Gaussian integration and normed algebras

Abstract

This paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. Our immediate motivation is a specific model, involving both boson and fermion fields, which arises as a representation of the continuous-time weakly self-avoiding walk. In this paper, we define normed algebras suitable for a renormalisation group analysis, and develop methods for performing analysis on these algebras. We also develop the theory of Gaussian integration on these normed algebras, and prove estimates for Gaussian integrals. The concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series.

arXiv comment43 pages

References in corpus (10)

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• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
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• Critical two-point function of the 4-dimensional weakly self-avoiding walk
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• Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
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• A renormalisation group method. IV. Stability analysis
• A renormalisation group method. V. A single renormalisation group step
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Cited by in corpus (21)

• 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
• The $1/N$ expansion of the symmetric traceless and the antisymmetric tensor models in rank three
• 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. V. A single renormalisation group step
• 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
• Random Polymers and Generalized Urn Processes
• Spectral gap critical exponent for Glauber dynamics of hierarchical spin models
• Four-dimensional weakly self-avoiding walk with contact self-attraction
• 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
• Energy of the Interacting Self-Avoiding Walk at the $\theta -$point
• Percolation transition for random forests in $d\ge 3$
• Duality of O(N) and Sp(N) random tensor models: tensors with symmetries