1995
DOI: 10.1007/bf02179382
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Instability of renormalization-group pathologies under decimation

Abstract: We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely deca… Show more

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Cited by 17 publications
(9 citation statements)
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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%
“…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%
“…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%
“…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%