Lectures on the renormalisation group

Brydges, David C.

doi:10.1090/pcms/016/02

Cited by 49 publications

(128 citation statements)

References 47 publications

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“…where m > 0 and ∇ is the discrete gradient operator acting on fields φ : Z d → R. In the case when V (z) = |z| 2 /2 + a d j=1 cos z j , z ∈ R d , the probability measure (1.2) describes the dual representation of a gas of lattice dipoles with activity a (see [3]). Our estimates on fluctuations will be uniform for m > 0.…”

Section: Introductionmentioning

confidence: 99%

A Strong Central Limit Theorem for a Class of Random Surfaces

Conlon

Spencer

2013

Commun. Math. Phys.

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Abstract. This paper is concerned with d = 2 dimensional lattice field models with action V (∇φ(·)), where V : R d → R is a uniformly convex function. The fluctuations of the variable φ(0) − φ(x) are studied for large |x| via the generating function given by g(x, µ) = ln e µ(φ(0)−φ(x)) A . In two dimensions g ′′ (x, µ) = ∂ 2 g(x, µ)/∂µ 2 is proportional to ln |x|. The main result of this paper is a bound on g ′′′ (x, µ) = ∂ 3 g(x, µ)/∂µ 3 which is uniform in |x| for a class of convex V . The proof uses integration by parts following Helffer-Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.

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Section: Introductionmentioning

confidence: 99%

A Strong Central Limit Theorem for a Class of Random Surfaces

Conlon

Spencer

2013

Commun. Math. Phys.

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“…We now claim that from (53) we obtain (55): If we smuggle in (55) the weight |x − y| α 2 q and apply Hölder's inequality first in x and then in y, we get 6 To show Sobolev's inequality in the outer domain {|x| > R} we argue as follows: By scale invariance, we may reduce ourselves to the domain {|x| > 1}; moreover, by standard approximation, we may assume u to be smooth and zero outside a ball big enough. We now extend u inside {|x| < 1} using the radial reflection x → x |x| 2 , apply Sobolev's inequality on the whole space and conclude by observing that, due to our choice of extension, the Dirichlet integral in {|x| < 1} can be controlled by the Dirichlet integral in {|x| > 1}.…”

Section: Proof Of Theoremmentioning

confidence: 94%

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

2017

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This paper is divided into two parts: In the main deterministic part, we prove that for an open domain D ⊂ R d with d ≥ 2, for every (measurable) uniformly elliptic tensor field a and for almost every point y ∈ D, there exists a unique Green's function centred in y associated to the vectorial operator −∇ · a∇ in D. This result implies the existence of the fundamental solution for elliptic systems when d > 2, i.e. the Green function for −∇ · a∇ in R d . In the second part, we introduce a shift-invariant ensemble · over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for |G(·; x, y)| , |∇ x G(·; x, y)| and |∇ x ∇ y G(·; x, y)| . These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case. Mathematics Subject Classification

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“…A different approach that is related to the scale-invariance of Z d is the renormalization group method, which has been very useful for several statistical physics models at criticality; see [Bry07]. We will not discuss this approach here.…”

Section: Percolation Renormalization and Scale-invariant Tilingsmentioning

confidence: 99%

Scale-invariant groups

Nekrashevych¹,

Pete²

2011

Groups Geom. Dyn.

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Abstract. Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F o Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS.1; m/; the affine groups A Ë Z d , for any A Ä GL.Z; d /. However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.Mathematics Subject Classification (2010). 20E08, 20F18, 05B45, 60B99, 60K35.

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Lectures on the renormalisation group

Cited by 49 publications

References 47 publications

A Strong Central Limit Theorem for a Class of Random Surfaces

A Strong Central Limit Theorem for a Class of Random Surfaces

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

Scale-invariant groups

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