David C. Brydges’s research while affiliated with University of British Columbia and other places

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the n-component model on when n=1,2, and prove that the critical Green's function is asymptotically a multiple of when and the coupling is weak. As another application of our method, we establish the analogous result for the lattice Edwards model at weak coupling. © 2021 Wiley Periodicals LLC.

We analyse the perturbative renormalisation group flow derived in the previous chapter. When nonperturbative effects are ignored, we show how this analysis leads to a logarithmic correction to mean-field scaling for the 4-dimensional hierarchical susceptibility. Nonperturbative effects however cannot be ignored, and we state the extension of the results for the perturbative flow to its nonperturbative version and use this extension to prove the main results for the |φ|⁴ model. The analysis of the nonperturbative effects is carried out in subsequent chapters. The notion of infrared asymptotic freedom is discussed.

We prove the main estimates on the nonperturbative coordinate of the renormalisation group map. This is the heart of the proof of the main result. One ingredient in the proof of the main estimates consists of the stability estimates which exploit the decay arising from $e^{-g|\varphi |{ }^4}$. A central ingredient in the proof consists of the crucial contraction which is responsible for the contractive property of the renormalisation group map. The contractive property was used in an essential way to construct the global renormalisation group flow in Chap. 6.

We give an introduction to some of the modifications needed to extend the renormalisation group method from the hierarchical to the Euclidean setting, and point out where in the literature these extensions can be found in full detail. Although the renormalisation group philosophy remains the same for the hierarchical and Euclidean models, the Euclidean setting requires significant adjustments. These include: an extended version of the perturbative coordinate, introduction of the circle product, analysis of field gradients within blocks, and more sophisticated norms involving regulators. We discuss all these items. The transfer of relevant contributions from the nonperturbative coordinate to the perturbative coordinate, in order to ensure the crucial contraction of the renormalisation group map, is now carried out via a sophisticated change of variables procedure. We also discuss this change of variables.

We define a hierarchical Gaussian field in a way that is motivated by the finite-range decomposition of the Gaussian free field. The hierarchical Gaussian free field is a hierarchical field that has comparable large distance behaviour to the lattice Gaussian free field. We explicitly construct a version of the hierarchical Gaussian field and verify that it has the desired properties. We define the hierarchical |φ|⁴ model and state the main result proved in this book, which gives the critical behaviour of the susceptibility of the 4-dimensional hierarchical |φ|⁴ model. In preparation for the proof of the main result, we reformulate the hierarchical |φ|⁴ model as a perturbation of a Gaussian integral.

We introduce the Tz-seminorm, which is used in subsequent chapters to measure the size of the nonperturbative coordinate of the renormalisation group map. We define the seminorm, prove its important product property, show how it can be used to obtain bounds on derivatives, and explain in which sense the seminorm of a Gaussian expectation is bounded by the expectation of the seminorm. Good properties of the seminorm with respect to exponentiation and Taylor expansion are developed; the latter is an essential ingredient in our proof of the crucial contraction property of the renormalisation group map. We conclude with some estimates on polynomials for later use.

We introduce the renormalisation group map for the hierarchical model. The renormalisation group map has two components: a perturbative and a nonperturbative coordinate. The analysis of the renormalisation group map occupies the remainder of the book. An advantage of the hierarchical model is that the analysis can be reduced to individual blocks; this is not the case in the Euclidean setting. We explain this reduction. We discuss progressive integration; irrelevant, marginal, and relevant polynomials; and localisation. The renormalisation group map involves the notion of flow of coupling constants (uj, gj, νj), as well as the flow of an infinite-dimensional non-perturbative coordinate Kj. The flow of coupling constants is given to leading order by perturbation theory, which is derived in this chapter.

We provide solutions to all the exercises given throughout the book.

We provide an introduction to the theory of critical phenomena and discusses several of the models which serve as guiding examples. The Ising and multi-component |φ|⁴ spin models are introduced and motivated, with emphasis on their critical behaviour. The theory of the mean-field model is developed in a self-contained manner. The Gaussian free field is introduced and its relation to simple random walk is explained. The notion of universality is discussed. Recent results for the critical behaviour of the |φ|⁴ model are summarised, including the existence of logarithmic corrections to mean-field critical exponents in dimension d = 4.

Following a brief discussion of the critical behaviour of the standard self-avoiding walk, we introduce the continuous-time weakly self-avoiding walk (also called the lattice Edwards model). We derive the BFS-Dynkin isomorphism which provides a random walk representation for spin systems. We introduce an anti-commuting fermion field represented by differential 1-forms, and explain an important connection with supersymmetry. We prove the localisation theorem, and use it to derive a representation of the weakly self-avoiding walk in terms of a supersymmetric spin system. For the 4-dimensional weakly self-avoiding walk, the renormalisation group method developed in this book has been extended to analyse the supersymmetric spin system and thereby yield results concerning the critical behaviour of the weakly self-avoiding walk.