Instability of renormalization-group pathologies under decimation

Martinelli, Fabio; Olivieri, E.

doi:10.1007/bf02179382

Cited by 17 publications

(9 citation statements)

References 11 publications

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“…Rivasseau [17] used perturbative and constructive renormalization to investigate rigorously the phenomenon of asymptotic freedom. Martinelli and Olivieri [14,15] investigated the stability and instability of pathologies of RG transformations under decimation. Haller and Kennedy [7] showed that a single RG transformation could map an area including a critical point to a set of well-defined renormalized interactions.…”

Section: Introductionmentioning

confidence: 99%

Renormalization group transformations near the critical point: Some rigorous results

Yin

2011

Journal of Mathematical Physics

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We consider renormalization group (RG) transformations for classical Ising-type lattice spin systems in the infinite-volume limit. Formally, the RG maps a Hamiltonian H into a renormalized Hamiltonian H ′ :With the help of the Dobrushin uniqueness condition and standard results on the polymer expansion, Haller and Kennedy gave a sufficient condition for the existence of the renormalized Hamiltonian in a neighborhood of the critical point. By a more complicated but reasonably straightforward application of the cluster expansion machinery, the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction. This in turn gives an upper bound for the RG linearization.

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Section: Introductionmentioning

confidence: 99%

Renormalization group transformations near the critical point: Some rigorous results

Yin

2011

Journal of Mathematical Physics

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“…Kashapov [9] worked with cumulants (semi-invariants), with estimates that relied on combinatorial methods of Malyshev [13], and showed that the RG map can be formalized rigorously in terms of the Hamiltonian of a Gibbs field. Martinelli and Olivieri [14] , [15] investigated the stability and instability of pathologies of RG transformations under decimation. For block-average RG transformations, Cammarota [3] proved that the block spin interaction tends in norm to a one-body quadratic potential in the infinite volume limit at high temperature, whereas van Enter [20] constructed a counterexample showing that the renormalized measure may not be Gibbsian even when the temperature is above the critical temperature.…”

Section: Introductionmentioning

confidence: 99%

A cluster expansion approach to renormalization group transformations

Yin

2011

Journal of Mathematical Physics

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The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the rigorous definition of the RG map for classical Ising-type lattice systems in the infinite volume limit at high temperature. A cluster expansion is used to justify the existence of the partial derivatives of the renormalized interaction with respect to the original interaction. This expansion is derived from the formal expressions, but it is itself well-defined and convergent. Suppose in addition that the original interaction is finite-range and translation-invariant. We will show that the matrix of partial derivatives in this case displays an approximate band property. This in turn gives an upper bound for the RG linearization.Comment: 13 page

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“…The above behavior is related to the fact that, given suitable values of the parameters β, h (close to the coexistence line h = 0, β > β c ), on a suitable scale ℓ some constrained models can undergo a phase transition (somehow related to the phase transition of the object system); whereas, given the same h, β, for sufficiently large scale ℓ any constrained model is in the weak coupling region. Another notion of robustness of the pathology is related to the application of decimation transformations, see [LV], [MO5].…”

Section: Introductionmentioning

confidence: 99%

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Bertini

Cirillo²,

Olivieri

1999

Journal of Statistical Physics

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In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite range lattice gases, when suitable strong mixing conditions are satisfied. Using block decimation procedure, cluster expansion (like in [HK]) and detailed comparison between statistical ensembles, we are able to prove Gibbsianess and convergence to a trivial (i.e. Gaussian and product) fixed point. Our results apply to 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.

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Instability of renormalization-group pathologies under decimation

Cited by 17 publications

References 11 publications

Renormalization group transformations near the critical point: Some rigorous results

Renormalization group transformations near the critical point: Some rigorous results

A cluster expansion approach to renormalization group transformations

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