Mean-Field Theory and the Gaussian Approximation

Kopietz, Peter; Bartosch, Lorenz; Schütz, Florian

doi:10.1007/978-3-642-05094-7_2

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…The scale separation avoids having an illconditioned gradient descent. The Ising approximation with an l 2 energy vector for p = 2 amounts to compute a Gaussian approximation of Ising, which is not precise, when we are close to the critical temperature [28]. One can indeed visualize important differences with the statistical distribution of original Ising in Figure 3(a).…”

Section: Renormalization and Waveletsmentioning

confidence: 99%

“…The Ising model has a phase transition at the critical temperature T c ≈ 2.27, from an 'ordered' to a 'disordered' state. The spin spatial correlation exhibits a characteristic scale ξ(T ) for T > T c and E{X(u)X(u + r)} e −|r|/ξ(T ) [28], with ξ(T c ) = 0. The correlation is self-similar at T = T c and E{X(u)X(u + r)} |r| −1/2 .…”

Section: Ising Processesmentioning

confidence: 99%

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Multiscale Sparse Microcanonical Models

Bruna¹,

Mallat²

2018

Preprint

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We study approximations of non-Gaussian stationary processes having long range correlations with microcanonical models. These models are conditioned by the empirical value of an energy vector, evaluated on a single realization. Asymptotic properties of maximum entropy microcanonical and macrocanonical processes and their convergence to Gibbs measures are reviewed. We show that the Jacobian of the energy vector controls the entropy rate of microcanonical processes.Sampling maximum entropy processes through MCMC algorithms require too many operations when the number of constraints is large. We define microcanonical gradient descent processes by transporting a maximum entropy measure with a gradient descent algorithm which enforces the energy conditions. Convergence and symmetries are analyzed. Approximations of non-Gaussian processes with long range interactions are defined with multiscale energy vectors computed with wavelet and scattering transforms. Sparsity properties are captured with l 1 norms. Approximations of Gaussian, Ising and point processes are studied, as well as image and audio texture synthesis.

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Section: Renormalization and Waveletsmentioning

confidence: 99%

Section: Ising Processesmentioning

confidence: 99%

Multiscale Sparse Microcanonical Models

Bruna¹,

Mallat²

2018

Preprint

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“…In quantum field theory, connected Green's functions are multipartite connected correlators of field operators [21]. Mean field theory is an approximation in which it is assumed that all connected correlators vanish [22]; in fact, mean field theory fails when there exist significant connected correlations, and one must then seek higher-order approximations. The cumulant expansion technique is similar to mean field theory, but only multipartite connected correlators of high enough order are ignored.…”

Section: Connected Correlationsmentioning

confidence: 99%

Lieb-Robinson bounds on n -partite connected correlation functions

et al. 2017

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Lieb and Robinson provided bounds on how fast bipartite connected correlations can arise in systems with only short-range interactions. We generalize Lieb-Robinson bounds on bipartite connected correlators to multipartite connected correlators. The bounds imply that an n-partite connected correlator can reach unit value in constant time. Remarkably, the bounds also allow for an n-partite connected correlator to reach a value that is exponentially large with system size in constant time, a feature which stands in contrast to bipartite connected correlations. We provide explicit examples of such systems.

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“…Problem 2. The random process is not absolutely suitable for the analysis of the virus prevalence on hierarchical architectures [10]. How should we describe the prevalence?…”

Section: Network Model and Preliminariesmentioning

confidence: 99%

Epidemics on Tree-Based Communities of Wireless Sensor Networks

Li¹

2016

Proceedings of the Fourth International Conference on Information Science and Cloud Computing — PoS(ISCC2015)

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Mean-Field Theory and the Gaussian Approximation

Cited by 6 publications

References 7 publications

Multiscale Sparse Microcanonical Models

Multiscale Sparse Microcanonical Models

Lieb-Robinson bounds on n -partite connected correlation functions

Epidemics on Tree-Based Communities of Wireless Sensor Networks

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