Introduction to a renormalisation group method

Abstract

This book provides an introduction to a renormalisation group method in the spirit of that of Wilson. It starts with a concise overview of the theory of critical phenomena and the introduction of several tools required in the renormalisation group approach, including Gaussian integration and finite range decomposition. The bulk of the book consists of an analysis of the 4-dimensional hierarchical $|\phi {|}^4$ model. This includes definition of the renormalisation group map, perturbation theory, development of a general family of norms that we use to analysis the renormalisation group map, and of a stability analysis of the dynamical system that the renormalisation group map defines. The analysis of the hierarchical $|\phi {|}^4$ model is presented in a pedagodical way and set up in such that the extension to the more complicated Euclidean nearest-neighbour setting is very much parallel. This extension is outlined in an appendix with precise pointers to the literature where it is carried out in detail. Finally, we provide a short introduction to supersymmetry and the supersymmetric representation of self-avoiding walks. The renormalisation group method can be extended to this supersymmetric setting and has been used to study the 4-dimensional weakly self-avoiding walk.

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References in corpus (22)

• Two Dimensional Gauge Theories Revisited
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Random Currents and Continuity of Ising Model's Spontaneous Magnetization
• High-precision estimate of the hydrodynamic radius for self-avoiding walks
• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Finite range Decomposition of Gaussian Processes
• Self-avoiding walk is sub-ballistic
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• A Complete Renormalization Group Trajectory Between Two Fixed Points
• Height fluctuations in interacting dimers
• A renormalisation group method. II. Approximation by local polynomials
• Critical exponents for long-range O(n) models below the upper critical dimension
• 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
• A renormalisation group method. IV. Stability analysis
• 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
• Localization for a nonlinear sigma model in a strip related to vertex reinforced jump processes
• Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps
• Finite Range Decomposition for Gaussian Measures with Improved Regularity
• On A Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus

Cited by in corpus (25)

• Log-Sobolev inequality for the continuum sine-Gordon model
• Equation of Motion Method to strongly correlated Fermi systems and Extended RPA approaches
• Gentle introduction to rigorous Renormalization Group: a worked fermionic example
• Self-avoiding walk, spin systems, and renormalization
• Log-Sobolev inequality for near critical Ising models
• Stochastic dynamics and the Polchinski equation: an introduction
• On the derivation of mean-field percolation critical exponents from the triangle condition
• A Lorentzian renormalisation group equation for gauge theories
• Mean-field tricritical polymers
• An Algebraic and Microlocal Approach to the Stochastic Non-linear Schrödinger Equation
• The geometry of random walk isomorphism theorems
• Loop-erased random walk as a spin system observable
• The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Spectral gap in mean-field $\mathcal{O}(n)$-model
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• Existence of a tricritical point for the Blume-Capel model on ${\mathbb{Z}}^d$
• The continuous-time lace expansion
• Universality in the 2d quasi-periodic Ising model and Harris-Luck irrelevance
• Percolation transition for random forests in $d\ge 3$
• One-matrix differential reformulation of two-matrix models
• On the running and the UV limit of Wilsonian renormalization group flows
• Large charge fluctuations in the hierarchical Coulomb gas
• Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed