End-to-End Distance from the Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

Brydges, David C.

doi:10.1007/s00220-003-0885-6

Cited by 23 publications

(39 citation statements)

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“…The proofs of all these results for the 4-dimensional hierarchical lattice are based on renormalisation group methods, but very different approaches are used in [3,5,6] and in [21]. The approach of [21] is based on a direct analysis of the self-avoiding paths themselves.…”

Section: Dimension D =mentioning

confidence: 99%

“…The approach of [21] is based on a direct analysis of the self-avoiding paths themselves. In contrast, the approach of [3,5,6], as well as the proof of Theorem 7.1, are based on a functional integral representation for the two-point function with no direct path analysis.…”

Section: Dimension D =mentioning

confidence: 99%

“…The point of departure of the proofs of Theorems 7.1-7.2, and more generally of the analysis in [3,5,6,8,9,16,43], is a functional integral representation for self-avoiding walks. Such representations have their roots in [45,42,38,3] and recently have been summarised and extended in [7].…”

Section: Functional Integral Representationmentioning

confidence: 99%

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The self-avoiding walk: A brief survey

Slade

2011

EMS Series of Congress Reports

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Abstract. Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more difficult to analyse and many of the important mathematical problems remain unsolved. This paper provides an overview of some of what is known about the critical behaviour of the self-avoiding walk, including some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics.2010 Mathematics Subject Classification. Primary 60K35, 82B41. Keywords. Self-avoiding walks, critical exponents, renormalisation group, lace expansion. Self-avoiding walksThis article provides an overview of the critical behaviour of the self-avoiding walk model on Z d , and in particular discusses how this behaviour differs as the dimension d is varied. The books [29,40] are general references for the model. Our emphasis will be on dimensions d = 4 and d ≥ 5, where results have been obtained using the renormalisation group and the lace expansion, respectively.An n-step self-avoiding walk from x ∈ Z d to y ∈ Z d is a map ω : {0, 1, . . . , n} → Z d with: ω(0) = x, ω(n) = y, |ω(i + 1) − ω(i)| = 1 (Euclidean norm), and ω(i) = ω(j) for all i = j. The last of these conditions is what makes the walk self-avoiding, and the second last restricts our attention to walks taking nearestneighbour steps.Let d ≥ 1. Let S n (x) be the set of n-step self-avoiding walks on Z d from 0 to x. Let S n = ∪ x∈Z d S n (x). Let c n (x) = |S n (x)|, and let c n = x∈Z d c n (x) = |S n |. We declare all walks in S n to be equally likely: each has probability c −1 n . See Figure 1. We write E n for expectation with respect to this uniform measure on S n . What it is not:• It is not the so-called "true" or "myopic" self-avoiding walk, i.e., the stochastic process which at each step looks at its neighbours and chooses uniformly from those visited least often in the past -the two models have different critical behaviour (see [28,50] for recent progress on the "true" self-avoiding walk).• It is by no means Markovian.• It is not a stochastic process: the uniform measures on S n do not form a consistent family.

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Section: Dimension D =mentioning

confidence: 99%

Section: Dimension D =mentioning

confidence: 99%

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The self-avoiding walk: A brief survey

Slade

2011

EMS Series of Congress Reports

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“…Also, there is no decay as n → ∞ in (2.12) when γ = 1. Partial results for the 4-dimensional case have been obtained in [43,44,129] (physics references include [38,65]). The 3-dimensional case is completely unsolved mathematically.…”

Section: (2d)mentioning

confidence: 99%

“…Partial results for d = 4 have been obtained in [43,44,129]. For d = 2, 3, 4, for the nearest-neighbour model with λ = 1, it is still an open problem even to prove the "obvious" bounds that the mean-square displacement is bounded below by n (cf.…”

Section: (2d)mentioning

confidence: 99%

The Lace Expansion and its Applications

Saint-Flour

Slade

Picard³

2006

Lecture Notes in Mathematics

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ii The Lace Expansion and its ApplicationsThe Lace Expansion and its Applications iii Preface Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. Although these models have been studied by mathematicians for about half a century, exciting new developments continue to occur and the subject is flourishing. Much progress has been made, but it remains a major challenge for mathematical physics and probability theory to obtain a complete and mathematically rigorous understanding of the scaling theory of these models at criticality.These lecture notes concern the lace expansion, which is a powerful tool for the analysis of the critical scaling of several models above their upper critical dimensions, namely:• the self-avoiding walk on Z d for d > 4, • lattice trees and lattice animals onResults include proofs of existence of critical exponents, with mean-field values, and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion. There are two distinct goals for these notes. The first goal is to provide a written accompaniment to my lectures at the Saint-Flour summer school in 2004, and at the Pacific Institute for the Mathematical Sciences -University of British Columbia summer school in 2005. The notes contain an introduction to the lace expansion and several of its applications, with sufficient background and depth to prepare a newcomer to do research using the lace expansion. Basic graduate level probability theory will be used, but no previous knowledge of the lace expansion or super-Brownian motion is assumed. The second goal is to provide a survey of the field, so that an interested reader can follow up by consulting the original literature. In pursuit of the second goal, these notes include more material than can be covered during a summer school course.Following a brief initial section concerning random walk, the notes can be divided into four parts, whose contents are summarized as follows.Part I, which concerns the self-avoiding walk, consists of Sections 2-6. A complete and self-contained proof is given of the convergence of the lace expansion for the nearest-neighbour model in dimensions d 4, and for the spread-out model of self-avoiding walks which take steps of length at most L, with L 1, in dimensions d > 4. The convergence proof presented here seems simpler than all previous lace expansion convergence proofs. As a consequence of convergence, it is shown that the critical exponent γ for the generating function of the number of n-step self-avoiding walks exists and is equal to 1. A survey is then given of the many extensions of this result that have been obtained using the lace expansion. Part II, which concerns lattice trees and lattice animals, consists of Sections 7-8. It is shown how a minor modification of the expansion for the self-avoiding walk can be applied to give expansions for lattice trees and lattice animals, and an indicatio...

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Polygons and the Lace Expansion

Clisby¹,

Slade

2009

Polygons, Polyominoes and Polycubes

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End-to-End Distance from the Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

Cited by 23 publications

References 10 publications

The self-avoiding walk: A brief survey

The self-avoiding walk: A brief survey

The Lace Expansion and its Applications

Polygons and the Lace Expansion

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