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 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.

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 the specific norms used to analyse the renormalisation group map, and specify the domain of the map. The choice of norms is based on considerations concerning the typical sizes of the fluctuation and block-spin fields. We state the main estimates on the renormalisation group map in two theorems, and then use these theorems to construct the global renormalisation group flow in the nonperturbative case. The construction requires, in particular, the construction of the critical point. The latter is done via the Bleher–Sinai argument. The results of this chapter reduce our analysis of the 4-dimensional hierarchical model to the proof of the estimates on the renormalisation group map stated in this chapter. The two theorems which state those estimates are proved in the next two chapters.

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.

Our implementation of the renormalisation group method relies on a finite-range decomposition of the Gaussian free field to allow progressive integration over scales. This requires an appropriate decomposition of the covariance of the Gaussian free field into a sum of simpler covariances. In this chapter, we provide a self-contained derivation of a finite-range covariance decomposition. This is easy for the case of the continuum, which we consider first. We then consider the lattice case, where the finite-range decomposition is generated by making use of the finite speed of propagation of the discrete wave equation. This then gives rise to a finite-range decomposition on the discrete torus. The finite-range decomposition provides our main motivation for the definition of the hierarchical model in Chap. 4, which is the focus of the book until Chap. 12 where Euclidean models are discussed.

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 provide solutions to all the exercises given throughout the book.

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.