Sine-Gordon Revisited

Dimock, J.; Hurd, T. R.

doi:10.1007/s000230050005

Cited by 42 publications

(68 citation statements)

References 27 publications

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“…One still expects the theory to be renormalisable until β 2 = 8π, which corresponds to Φ 3 6 , Φ 4 4 and KPZ in dimension 2, at which point renormalisability breaks down. This suggests that the value β 2 = 8π is critical for (1.1) and that there is no hope to give it any non-trivial meaning beyond that, see for example [DH00,Fal12] and, in a slightly different context, [LRV13]. This heuristic (including the fact that Wick renormalisability breaks down at β 2 = 4π) is well-known and has been demonstrated in [Frö76,BGN82,Nic83,NRS86,DH00] at the level of the behaviour of the partition function for the corresponding lattice model.…”

Section: Introductionmentioning

confidence: 99%

The Dynamical Sine-Gordon Model

Hairer

Shen

2015

Commun. Math. Phys.

150

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We introduce the dynamical sine-Gordon equation in two space dimensions with parameter β, which is the natural dynamic associated to the usual quantum sineGordon model. It is shown that when β 2 ∈ (0, 16π 3 ) the Wick renormalised equation is well-posed. In the regime β 2 ∈ (0, 4π), the Da Prato-Debussche method [DPD02, DPD03] applies, while for3 ), the solution theory is provided via the theory of regularity structures [Hai13]. We also show that this model arises naturally from a class of 2+1-dimensional equilibrium interface fluctuation models with periodic nonlinearities.The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in [LRV13].

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Section: Introductionmentioning

confidence: 99%

The Dynamical Sine-Gordon Model

Hairer

Shen

2015

Commun. Math. Phys.

150

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“…Our particular line of attack is influenced by a series of works which started with [BY], developed further in [DH1,DH2,, with more recent developments in [MS]. We shall be concerned here with these latter developments.…”

Section: Introductionmentioning

confidence: 99%

Critical (?4)3,?

Brydges

Mitter

Scoppola

2003

Communications in Mathematical Physics

127

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The Euclidean (φ 4 ) 3, ǫ model in R 3 corresponds to a perturbation by a φ 4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0 ≤ ε ≤ 1. For ε = 1 one recovers the covariance of a massless scalar field in R 3 . For ε = 0 φ 4 is a marginal interaction. For 0 ≤ ε < 1 the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for ε > 0, sufficiently small, there exists a non-gaussian fixed point ( with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed.

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“…The construction of [GK85] is particularly well suited to be used as the input of a single cluster expansion. While the more recent implementation of the renormalization group by Brydges and collaborators ([BY90, BDH95, BS15, AKM16]) has been developed in more detail and also been applied to critical phenomena of other interesting models (such as long range models below the critical dimension [BMS03,Sla18,LSW17], or the dipole gas [DH00,Fal13]), and while this method shares the intuitive inductive nature of the [GK85] argument and uses similar expansions, there are obstacles to using it as the basis of the current paper. In particular, the behavior of effective interactions at large fields is not tracked as precisely as in [GK85], and currently our method relies on such precise large field bounds.…”

Section: Introductionmentioning

confidence: 99%

Critical Exponent for the Magnetization of the Weakly Coupled $$\phi _4^4 $$ Model

Lohmann

2020

Commun. Math. Phys.

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We consider the weakly coupled φ 4 theory on Z 4 , in a weak magnetic field h, and at the chemical potential νc for which the theory is critical if h = 0. We prove that, as h ց 0, the magnetization of the model behaves as (h log h −1 ) 1 3 , and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to physicists, but had never been proven rigorously. Our proof uses the classic construction of the critical theory by Gawedzki and Kupiainen [GK85], and a cluster expansion with large blocks.

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Sine-Gordon Revisited

Cited by 42 publications

References 27 publications

The Dynamical Sine-Gordon Model

The Dynamical Sine-Gordon Model

Critical (?4)3,?

Critical Exponent for the Magnetization of the Weakly Coupled $$\phi _4^4 $$ Model

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