# Gordon Slade

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - Nov 2019)

The central concern of this thesis is the study of critical behaviour in models of statistical physics in the upper-critical dimension. We study a generalized n-component lattice |φ|⁴ model and a model of weakly self-avoiding walk with nearest-neighbour contact self-attraction on the Euclidean lattice ℤd. By utilizing a supersymmetric integral representation involving boson and fermion fields, the two models are studied in a unified manner.Our main result, which is contingent on a small coupling hypothesis, identifies the precise leading-order asymptotics of the two-point function, susceptibility, and finite-order correlation length of both models in d = 4. In particular, we show that the critical two-point function satisfies mean-field scaling whereas the near-critical susceptibility and finite-order correlation length exhibit logarithmic corrections to mean-field behaviour. The proof employs a renormalization group method of Bauerschmidt, Brydges, and Slade based on a finite-range covariance decomposition and requires two extensions to this method.The first extension, which is required for the computation of the finite-order correlation length (even for the ordinary weakly self-avoiding walk and |φ|⁴ model), is an improvement of the norms used to control the evolution of the renormalization group. This allows us to obtain improved error estimates in the massive regime of the renormalization group flow.The second extension involves the identification of critical parameters for models initialized with a non-zero error coordinate coupled to a marginal/relevant coordinate. This allows us, for example, to realize the two-point function and susceptibility for the walk with self-attraction as a small perturbation of the corresponding quantities without self-attraction, whose asymptotic behaviour was determined by Bauerschmidt, Brydges, and Slade. This establishes a form of universality.

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Critical phenomena and phase transitions are important subjects in statistical mechanics and probability theory. They are connected to the phenomenon of universality that makes the study of mathematically simple models physically relevant. Examples of such models include ferromagnetic spin systems such as the Ising, O(n) and n-component |φ|⁴ models, but also the self-avoiding walk that has been observed to formally correspond to a "zero-component" spin model by de Gennes.Our subject in this thesis is the extension and application of a rigorous renormalisation group method developed by Bauerschmidt, Brydges and Sladeto study the critical behaviour of the continuous-time weakly self-avoiding walk and of the n-component |φ|⁴ model on the 4-dimensional square lattice ℤ⁴. Although a "zero-component" vector is mathematically undefined (at least naively), we are able to interpret the weakly self-avoiding walk in a mathematically rigorous manner as the n = 0 case of the n-component |φ|⁴ model, and provide a unified treatment of both models. For the |φ|⁴ model, we determine the asymptotic decay of the critical correlation functions including the logarithmic corrections to Gaussian scaling, for n ≥ 1. This extends previously known results for n = 1 to all n ≥ 1, and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually- and self-avoiding walks, for all p ≥ 1, including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting.

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The main results of this thesis concern the spatial decomposition of Gaussian fields and the structural stability of a class of dynamical systems near a non-hyperbolic fixed point. These are two problems that arise in a renormalization group method for random fields and self-avoiding walks developed by Brydges and Slade. This renormalization group program is outlined in the introduction of this thesis with emphasis on the relevance of the problems studied subsequently. The first original result is a new and simple method to decompose the Green functions corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green function. This result gives rise to multiscale decompositions of the associated free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Such decompositions are the point of departure for renormalization group analysis. The novelty of the result is the use of the finite propagation speed of the wave equation and a related property of Chebyshev polynomials. The result improves several existing results and also gives simpler proofs. The second result concerns structural stability, with respect to contractive third-order perturbations, of a certain class of dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well- posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove the existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. This result can be applied to the renormalization group map of Brydges and Slade, and is an ingredient in the analysis of the long-distance behavior of four dimensional weakly self-avoiding walks using this approach.

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We study lattice trees and lattice animals in high dimensions. Lattice trees and animals are interesting combinatorial objects used to model branched polymers in polymer science. They are also of interest in combinatorics and in the study of critical phenomena in statistical physics.[Abstract portion beginning here modified and differs from the print copy]. Our study takes place in the nearest-neighbor and spread-out models on the d-dimensional integer lattice. On either graph, a lattice animal is a finite connected subgraph, and a lattice tree is an animal without cycles. Let t_n and a_n be the number of lattice trees and animals with n bonds that contain the origin, respectively. Standard subadditivity arguments provide the existence of the growth constants τ and α, which are the limit, as the dimension d goes to infinity, of the n-th root of t_n and a_n, respectively.] We are interested in the critical points of these models, which are the reciprocals of the corresponding growth constants.We rigorously calculate the first three terms of a 1/d-expansion for the critical points of nearest-neighbor lattice trees and animals. The proof follows a recursive argument similar to the one used by Hara and Slade (1995), van der Hofstad and Slade (2006), to obtain analogous results for the critical points of self-avoiding walks and percolation. To provide the leading terms in the expansions, we use a mean-field model, related to the Galton-Watson branching process with critical Poisson offspring distribution, and results obtained with the lace expansion. The leading terms are also calculated in the spread-out model. Then we develop expansions for the nearest-neighbor generating functions and, together with the lace expansion, obtain the first and second correction terms.Our result gives a rigorous proof for previous work on the subject [11],[21, 36]. Given the algorithmic nature of the proof, it can be extended, with sufficient labor, to compute higher degree terms. It may provide the starting point for proving the existence of an asymptotic expansion with rational coefficients, for the critical point of nearest-neighbor lattice trees.

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Invasion percolation is an infinite subgraph of an infinite connected graph with finite degrees, defined inductively as follows. To each edge of the underlying graph, attach a random edge weight chosen uniformly from [0,1], independently for each edge. Starting from a single vertex, a cluster is grown by adding at each step the boundary edge with least weight. Continue this process forever to obtain the invasion cluster. In the following, we consider the case where the underlying graph is a regular tree: starting from the root, each vertex has a fixed number of children. In chapter 2, we study the structure of the invasion cluster, considered as a subgraph of the underlying tree. We show that it consists of a single backbone, the unique infinite path in the cluster, together with sub-critical percolation clusters emerging at every point along the backbone. By studying the scaling properties of the sub-critical parameters, we obtain detailed results such as scaling formulas for the r-point functions, limiting Laplace transforms for the level sizes and volumes within balls, and mutual singularity compared to the incipient infinite cluster.Chapter 3 gives the scaling limit of the invasion cluster. This is a random continuous tree described by a drifted Brownian motion, with a drift that depends on a certain local time. This representation also yields a probabilistic interpretation of the level size scaling limit.Finally, chapter 4 studies the internal structure of the invasion cluster through its ponds and outlets. These are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. Tail asymptotics for fixed ponds are also derived.

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##### Master's Student Supervision (2010 - 2018)

The barycentric subdivision dissects a triangle along its three medians into six childrentriangles. The children of a flat triangle (i.e. the vertices are collinear) are flat. For anytriangle Δ let the shape S(Δ) be the unique complex point z in the first quadrant such thatΔ is similar to the triangle −1, 1, z in which the edge between −1 and 1 has maximal length.Only flat triangles’ shapes lie on the real line. If Δ^(n) is a Markov chain of triangles with Δ^(n)chosen uniformly amongst the children of Δ^(n−1), then we call the Markov chain S(Δ^(n)) ashape chain and we call it (non-)flat, if Δ(⁰) and therefore each Δ^(n) is (non-)flat. Let Znbe a non-flat and Xn be a flat shape chain. We say that a sequence of random variables Wntaking values in ℂ \ {0} decays exactly or at least with rate X, if X > 0 and almost surelylimn1/n*ln|Wn| = −X or lim supn1/nln |Wn| ≤ −X, resp. In a paper from 2011, P. Diaconisand L. Miclo show that =Zn decays at least with rate X' for some universal constant X' andthat Xn has an invariant measure μ. We prove that =Zn decays exactly with rate X for auniversal constant X which we express as an integral w.r.t. μ. The above paper also showsthe convergence of Zn − Xn to 0 in probability for a specific coupling (Xn,Zn). For thiscoupling we prove that Zn − Xn decays exactly with rate X.

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