arXiv subjects

Gordon Slade

Publications and source records attributed to Gordon Slade.

At least 37 records · Page 2Linked to original sources

arXiv2020

Three-dimensional tricritical spins and polymers

We consider two intimately related statistical mechanical problems on $\mathbb{Z}^3$: (i) the tricritical behaviour of a model of classical unbounded $n$-component continuous spins with a triple-well single-spin potential (the $|φ|^6$ model), and (ii) a random walk model of linear polymers with a three-body repulsion and two-body attraction at the tricritical theta point (critical point for the collapse transition) where repulsion and attraction effectively cancel. The polymer model is exactly equivalent to a supersymmetric spin model which corresponds to the $n=0$ version of the $|φ|^6$ model. For the spin and polymer models, we identify the tricritical point, and prove that the tricritical two-point function has Gaussian long-distance decay, namely $|x|^{-1}$. The proof is based on an extension of a rigorous renormalisation group method that has been applied previously to analyse the $|φ|^4$ and weakly self-avoiding walk models on $\mathbb{Z}^4$.

math-ph↗

arXiv2019

Introduction to a renormalisation group method

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 $|φ|^4$ 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 $|φ|^4$ 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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arXiv2019

Self-avoiding walk on the complete graph

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 simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.

math.PR↗

arXiv2019

Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

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 order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.

math.PR↗

arXiv2018

Self-avoiding walk, spin systems, and renormalization

The self-avoiding walk, and lattice spin systems such as the $φ^4$ model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous nonperturbative renormalization group method.

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arXiv2017

Critical exponents for long-range O(n) models below the upper critical dimension

We consider the critical behaviour of long-range $O(n)$ models ($n \ge 0$) on ${\mathbb Z}^d$, with interaction that decays with distance $r$ as $r^{-(d+α)}$, for $α\in (0,2)$. For $n \ge 1$, we study the $n$-component $|φ|^4$ lattice spin model. For $n =0$, we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension $d_c=2α$. For dimensions $d=1,2,3$ and small $ε>0$, we choose $α= \frac 12 (d+ε)$, so that $d=d_c-ε$ is below the upper critical dimension. For small $ε$ and weak coupling, to order $ε$ we prove existence of and compute the values of the critical exponent $γ$ for the susceptibility (for $n \ge 0$) and the critical exponent $α_H$ for the specific heat (for $n \ge 1$). For the susceptibility, $γ= 1 + \frac{n+2}{n+8} \frac εα+ O(ε^2)$, and a similar result is proved for the specific heat. Expansion in $ε$ for such long-range models was first carried out in the physics literature in 1972. Our proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension $d=4$, and is based on the construction of a non-Gaussian renormalisation group fixed point. Some aspects of the method simplify below the upper critical dimension, while some require different treatment, and new ideas and techniques with potential future application are introduced.

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arXiv2017

Critical two-point function for long-range $O(n)$ models below the upper critical dimension

We consider the $n$-component $|φ|^4$ lattice spin model ($n \ge 1$) and the weakly self-avoiding walk ($n=0$) on $\mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+α)}$ with $α\in (0,2)$. The upper critical dimension is $d_c=2α$. For $ε>0$, and $α= \frac 12 (d+ε)$, the dimension $d=d_c-ε$ is below the upper critical dimension. For small $ε$, weak coupling, and all integers $n \ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-α)}$. This "sticking" of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.

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arXiv2017

Four-dimensional weakly self-avoiding walk with contact self-attraction

We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb{Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $p$ (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction.

math-ph↗

arXiv2016

Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|φ|^4$ spins

We study the 4-dimensional $n$-component $|φ|^4$ spin model for all integers $n \ge 1$, and the 4-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case $n=0$ interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order $p$, and prove the existence of a logarithmic correction to mean-field scaling, with power $\frac 12\frac{n+2}{n+8}$, for all $n \ge 0$ and $p>0$. The proof is based on an improvement of a rigorous renormalisation group method developed previously.

math-ph↗

arXiv2016

Renormalisation group analysis of 4D spin models and self-avoiding walk

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 prove that the susceptibility diverges with a logarithmic correction to the mean-field behaviour with exponent $(n+2)/(n+8)$. This result extends rigorously to $n=0$, interpreted as a supersymmetric version of the model that represents exactly the continuous-time weakly self-avoiding walk. We also analyse the critical two-point function of the weakly self-avoiding walk, the specific heat and pressure of the $|φ|^4$ model, as well as scaling limits of the spin field close to the critical point.

math-ph↗

arXiv2015

Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|φ|^4$ model

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice $\mathbb{Z}^4$, for the weakly coupled $n$-component $|φ|^4$ spin model for all $n \geq 1$, and for the continuous-time weakly self-avoiding walk. For the $|φ|^4$ model, we prove that the critical two-point function has $|x|^{-2}$ (Gaussian) decay asymptotically, for $n \ge 1$. We also determine the asymptotic decay of the critical correlations of the squares of components of $φ$, including the logarithmic corrections to Gaussian scaling, for $n \geq 1$. The above extends previously known results for $n = 1$ to all $n \ge 1$, 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 $p \ge 1$, 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. In addition, for both models, we study the approach to the critical point and prove existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the $n = 0$ case of the $|φ|^4$ model, and provides a unified treatment of both models, and of all the above results.

math-ph↗

arXiv2014

Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the critical dimension $d=4$, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent 1/4 for the logarithm. The susceptibility has been well understood previously for dimensions $d \geq 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

math-ph↗

arXiv2014

Critical two-point function of the 4-dimensional weakly self-avoiding walk

We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $η$ exists and is equal to zero. Results of this nature have been proved previously for dimensions $d \geq 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

math-ph↗

arXiv2014

A renormalisation group method. I. Gaussian integration and normed algebras

This paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. Our immediate motivation is a specific model, involving both boson and fermion fields, which arises as a representation of the continuous-time weakly self-avoiding walk. In this paper, we define normed algebras suitable for a renormalisation group analysis, and develop methods for performing analysis on these algebras. We also develop the theory of Gaussian integration on these normed algebras, and prove estimates for Gaussian integrals. The concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series.

math-ph↗

arXiv2014

A renormalisation group method. II. Approximation by local polynomials

This paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The method is set within a normed algebra $\mathcal{N}$ of functionals of the fields. In this paper, we develop a general method---localisation---to approximate an element of $\mathcal{N}$ by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to $F$ in $\mathcal{N}$ amounts to the extraction of the relevant and marginal parts of $F$. We prove estimates relating $F$ and its corresponding local polynomial, in terms of the $T_ϕ$ semi-norm introduced in part I of the series.

math-ph↗

arXiv2014

A renormalisation group method. III. Perturbative analysis

This paper is the third in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. In this paper, we motivate and present a general approach towards second-order perturbative renormalisation, and apply it to a specific supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$. Our focus is on the critical dimension $d=4$. The results include the derivation of the perturbative flow of the coupling constants, with accompanying estimates on the coefficients in the flow. These are essential results for subsequent application to the 4-dimensional weakly self-avoiding walk, including a proof of existence of logarithmic corrections to their critical scaling. With minor modifications, our results also apply to the 4-dimensional $n$-component $|φ|^4$ spin model.

math-ph↗

arXiv2014

A renormalisation group method. IV. Stability analysis

This paper is the fourth in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The third paper in the series presents a perturbative analysis of a supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$. We now present an analysis of the relevant interaction functional of the supersymmetric field theory, which permits a nonperturbative analysis to be carried out in the critical dimension $d = 4$. The results in this paper include: proof of stability of the interaction, estimates which enable control of Gaussian expectations involving both boson and fermion fields, estimates which bound the errors in the perturbative analysis, and a crucial contraction estimate to handle irrelevant directions in the flow of the renormalisation group. These results are essential for the analysis of the general renormalisation group step in the fifth paper in the series.

math-ph↗

arXiv2014

A renormalisation group method. V. A single renormalisation group step

This paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and/or fermion fields, and comprises the core of the method. In the renormalisation group method, increasingly large scales are studied in a progressive manner, with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map. In our context, the progressive analysis is performed via a finite-range covariance decomposition. Perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial, but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results. In this paper, we define an additional non-perturbative coordinate, which together with the flow of coupling constants defines the complete evolution of the renormalisation group map. We specify conditions under which the non-perturbative coordinate is contractive under a single renormalisation group step. Our framework is essentially combinatorial, but its implementation relies on analytic results developed earlier in the series of papers. The results of this paper are applied elsewhere to analyse the critical behaviour of the 4-dimensional continuous-time weakly self-avoiding walk and of the 4-dimensional $n$-component $|φ|^4$ model. In particular, the existence of a logarithmic correction to mean-field scaling for the susceptibility can be proved for both models, together with other facts about critical exponents and critical behaviour.

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