Tightness of the Ising–Kac Model on the Two-Dimensional Torus

Hairer, Martin; Iberti, Massimo

doi:10.1007/s10955-018-2033-x

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…Inspired by this result, our aim here is to show how these recent ideas connecting probability with PDE theory can be streamlined and extended to recover a complete, self-contained and simple, proof of existence of the Φ 4 3 measure on the full space. In the same spirit see also the work of Hairer and Iberti [HI18] on the tightness of the 2d Ising-Kac model. Soon after Hairer's seminal paper [Hai14], Jaffe [Jaf14] analyzed the stochastic quantization from the point of view of reflection positivity and constructive QFT and concluded that one has to necessarily take the infinite time limit to satisfy RP.…”

Section: Introductionmentioning

confidence: 80%

A PDE construction of the Euclidean $Φ^4_3$ quantum field theory

Massimiliano¹,

Hofmanová²

2018

Preprint

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We present a self-contained construction of the Euclidean Φ 4 quantum field theory on R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on R 3 defined on a periodic lattice of mesh size ε and side length M . We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as ε → 0, M → ∞. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder-Schrader axioms for a nontrivial Euclidean QFT apart from rotation invariance and clustering. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson-Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields. Our argument applies to arbitrary positive coupling constant and also to multicomponent models with O(N ) symmetry.

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Section: Introductionmentioning

confidence: 80%

A PDE construction of the Euclidean $Φ^4_3$ quantum field theory

Massimiliano¹,

Hofmanová²

2018

Preprint

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“…For a proof of restricted Markov uniqueness of dynamics associated with the Φ 4 2 -model see [87], which uses also results of [74], [74] providing also a new construction of strong solutions in certain negative index Besov spaces for this stochastic quantization equation. For a derivation of the stochastic quantization equation from Kac-Ising models see [35], [41], [55] and [72]. For work on Gaussian white noise driven PDEs related to other models of quantum fields in 2-dimensional space-time see [2], [9] and [59].…”

Section: The Construction Of a Corresponding φmentioning

confidence: 99%

The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model

Albeverio¹,

Kusuoka²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

235

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We give a direct construction of invariant measures and global flows for the stochastic quantization equation to the quantum field theoretical Φ 4 3 -model on the 3-dimensional torus. This stochastic equation belongs to a class of singular stochastic partial differential equations (SPDEs) presently intensively studied, especially after Hairer's groundbreaking work on regularity structures. Our direct construction exhibits invariant measures and flows as limits of the (unique) invariant measures for corresponding finite dimensional approximation equations. Our work is done in the setting of distributional Besov spaces, adapting semigroup techniques for solving nonlinear dissipative parabolic equations on such spaces and using methods that originated from work by Gubinelli et al on paracontrolled distributions for singular SPDEs.

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“…Let us also mention that a derivation of the SQE for the ϕ 4 2 -model on R 2 from a Kac-Ising model has been achieved in [146]. This goes back to work initiated in [79] for d = 1 and conjectured results in [90] for d = 2, 3 (see also [110]). The SQEs for the exponential/trigonometric models on R 2 have been discussed by Dirichlet forms in [18] and by semigroup methods, similarly as in [22] (the latter for the ϕ 4 3 -model on T 3 ).…”

Section: Introduction 1backgroundmentioning

confidence: 69%

Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

Albeverio¹,

Kusuoka²

2021

Preprint

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A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures µ associated with the ϕ 4 3 -model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the ϕ 4 3 -model. Our starting point is a suitable approximation µ M,N of the measure µ we intend to construct. µ M,N is parametrized by an M -dependent space cut-off function ρ M : R 3 → R and an N -dependent momentum cut-off function ψ N : R 3 ∼ = R 3 → R, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions (X M,N t , t ≥ 0) that have µ M,N as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes (X M,N t , t ≥ 0) M,N . Limit points in the sense of convergence in law exist, when both M and N diverge to +∞. The limit processes (X t ; t ≥ 0) are continuous on the intersection of suitable Besov spaces and any limit point µ of the µ M,N is a stationary measure of X. µ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that µ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.

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Tightness of the Ising–Kac Model on the Two-Dimensional Torus

Cited by 8 publications

References 17 publications

A PDE construction of the Euclidean $Φ^4_3$ quantum field theory

A PDE construction of the Euclidean $Φ^4_3$ quantum field theory

The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model

Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

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