Gordon Slade
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View article: Evaluation of a pro-recovery training intervention (REFOCUS-RETAFORM) in specialist mental health services across France: stepped-wedge cluster randomised controlled trial protocol
Evaluation of a pro-recovery training intervention (REFOCUS-RETAFORM) in specialist mental health services across France: stepped-wedge cluster randomised controlled trial protocol Open
Clinical Trials NCT05824234, registered 21 April 2023.
View article: Critical scaling profile for trees and connected subgraphs on the complete graph
Critical scaling profile for trees and connected subgraphs on the complete graph Open
We analyze generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of …
View article: Critical scaling profile for trees and connected subgraphs on the complete graph
Critical scaling profile for trees and connected subgraphs on the complete graph Open
We analyze generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of …
View article: Near-critical and finite-size scaling for high-dimensional lattice trees and animals
Near-critical and finite-size scaling for high-dimensional lattice trees and animals Open
We consider spread-out models of lattice trees and lattice animals on $\mathbb Z^d$, for $d$ above the upper critical dimension $d_{\mathrm c}=8$. We define a correlation length and prove that it diverges as $(p_c-p)^{-1/4}$ at the critica…
View article: Boundary conditions and the two-point function plateau for the hierarchical $|φ|^4$ model in dimensions 4 and higher
Boundary conditions and the two-point function plateau for the hierarchical $|φ|^4$ model in dimensions 4 and higher Open
We obtain precise plateau estimates for the two-point function of the finite-volume weakly-coupled hierarchical $|φ|^4$ model in dimensions $d \ge 4$, for both free and periodic boundary conditions, and for any number $n \ge 1$ of componen…
View article: Gaussian deconvolution and the lace expansion
Gaussian deconvolution and the lace expansion Open
We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = δ_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results…
View article: Gaussian deconvolution and the lace expansion for spread-out models
Gaussian deconvolution and the lace expansion for spread-out models Open
We present a new proof of $|x|^{-(d-2)}$ decay of critical two-point functions for spread-out statistical mechanical models on $\mathbb{Z}^d$ above the upper critical dimension, based on the lace expansion and assuming appropriate diagramm…
View article: Boundary conditions and universal finite-size scaling for the hierarchical $|φ|^4$ model in dimensions 4 and higher
Boundary conditions and universal finite-size scaling for the hierarchical $|φ|^4$ model in dimensions 4 and higher Open
We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|φ|^4$ model for all integers $n \ge 1$ in all dimensions $d\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that f…
View article: Weakly self-avoiding walk on a high-dimensional torus
Weakly self-avoiding walk on a high-dimensional torus Open
How long does a self-avoiding walk on a discrete $d$-dimensional torus have\nto be before it begins to behave differently from a self-avoiding walk on\n$\\mathbb{Z}^d$? We consider a version of this question for weakly self-avoiding\nwalk …
View article: Asymptotic behaviour of the lattice Green function
Asymptotic behaviour of the lattice Green function Open
The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long distance behaviour using the Laplace method applied to an integral represen…
View article: Self-avoiding walk on the hypercube
Self-avoiding walk on the hypercube Open
We study the number $c_n^{(N)}$ of $n$-step self-avoiding walks on the $N$-dimensional hypercube, and identify an $N$-dependent \emph{connective constant} $μ_N$ and amplitude $A_N$ such that $c_n^{(N)}$ is $O(μ_N^n)$ for all $n$ and $N$, a…
View article: Weakly self-avoiding walk on a high-dimensional torus
Weakly self-avoiding walk on a high-dimensional torus Open
How long does a self-avoiding walk on a discrete $d$-dimensional torus have to be before it begins to behave differently from a self-avoiding walk on $\mathbb{Z}^d$? We consider a version of this question for weakly self-avoiding walk on a…
View article: High-dimensional near-critical percolation and the torus plateau
High-dimensional near-critical percolation and the torus plateau Open
We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of tech…
View article: Asymptotic behaviour of the lattice Green function
Asymptotic behaviour of the lattice Green function Open
The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long-distance behaviour via a detailed analysis of an integral representation in…
View article: Mean-field tricritical polymers
Mean-field tricritical polymers Open
We provide an introductory account of a tricritical phase diagram, in the\nsetting of a mean-field random walk model of a polymer density transition, and\nclarify the nature of the density transition in this context. We consider a\ncontinu…
View article: The near-critical two-point function for weakly self-avoiding walk in high dimensions
The near-critical two-point function for weakly self-avoiding walk in high dimensions Open
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|…
View article: The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions
The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions Open
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|…
View article: Kotani's Theorem for the Fourier Transform
Kotani's Theorem for the Fourier Transform Open
In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function $f(x)$ on ${\mathbb Z}^d$ decays like $|x|^{-(d-2)}$ for large $x$, assuming that its Fourier transform $\hat f(k)$ is such that $|k|^{2}\h…
View article: Self-avoiding walk on the complete graph
Self-avoiding walk on the complete graph Open
There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the sim…
View article: Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees
Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees Open
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of…
View article: Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees
Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees Open
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of…
View article: Renormalisation group analysis of 4D spin models and self-avoiding walk
Renormalisation group analysis of 4D spin models and self-avoiding walk Open
We give an overview of results on critical phenomena in 4 dimensions, obtained recently using a rigorous renormalisation group method. In particular, for the $n$-component $|φ|^4$ spin model in dimension 4, with small coupling constant, we…