The paradox of The Netherlands

Why a successful economy is struggling?

A. van Witteloostuijn, C. Hendriks

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

Abstract

For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λc(ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree Δ when λ < λc(TΔ)λ < λc(TΔ) where TΔ is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ2 by proving that SSM holds on its self-avoiding walk tree Tsaw(ℤ2), and as a consequence he obtained that λc(Z2) ≥ λc(T4) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λc(ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on Tsaw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λc(ℤ2) > 2.48.
Original languageEnglish
Title of host publicationThe Netherlands as an EU Member
Subtitle of host publicationAwkward or Loyal Partner?
EditorsA. Schout, J. Rood
Place of PublicationThe Hague / Portland, OR
PublisherEleven International Publishing
Pages213-230
Number of pages312
ISBN (Print)9789490947996
Publication statusPublished - 2013

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Paradox
FPTAS
Lower bound
Lattice Gas Model
Independent Set
Maximum Degree
Partition Function
Refinement
Uniqueness
Upper bound
Integer
Graph in graph theory

Cite this

van Witteloostuijn, A., & Hendriks, C. (2013). The paradox of The Netherlands: Why a successful economy is struggling? In A. Schout, & J. Rood (Eds.), The Netherlands as an EU Member: Awkward or Loyal Partner? (pp. 213-230). The Hague / Portland, OR: Eleven International Publishing.
van Witteloostuijn, A. ; Hendriks, C. / The paradox of The Netherlands : Why a successful economy is struggling?. The Netherlands as an EU Member: Awkward or Loyal Partner?. editor / A. Schout ; J. Rood. The Hague / Portland, OR : Eleven International Publishing, 2013. pp. 213-230
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abstract = "For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λc(ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree Δ when λ < λc(TΔ)λ < λc(TΔ) where TΔ is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ2 by proving that SSM holds on its self-avoiding walk tree Tsaw(ℤ2), and as a consequence he obtained that λc(Z2) ≥ λc(T4) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λc(ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on Tsaw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λc(ℤ2) > 2.48.",
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van Witteloostuijn, A & Hendriks, C 2013, The paradox of The Netherlands: Why a successful economy is struggling? in A Schout & J Rood (eds), The Netherlands as an EU Member: Awkward or Loyal Partner?. Eleven International Publishing, The Hague / Portland, OR, pp. 213-230.

The paradox of The Netherlands : Why a successful economy is struggling? / van Witteloostuijn, A.; Hendriks, C.

The Netherlands as an EU Member: Awkward or Loyal Partner?. ed. / A. Schout; J. Rood. The Hague / Portland, OR : Eleven International Publishing, 2013. p. 213-230.

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

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N2 - For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λc(ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree Δ when λ < λc(TΔ)λ < λc(TΔ) where TΔ is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ2 by proving that SSM holds on its self-avoiding walk tree Tsaw(ℤ2), and as a consequence he obtained that λc(Z2) ≥ λc(T4) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λc(ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on Tsaw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λc(ℤ2) > 2.48.

AB - For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λc(ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree Δ when λ < λc(TΔ)λ < λc(TΔ) where TΔ is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ2 by proving that SSM holds on its self-avoiding walk tree Tsaw(ℤ2), and as a consequence he obtained that λc(Z2) ≥ λc(T4) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ2 and obtained that λc(ℤ2) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on Tsaw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λc(ℤ2) > 2.48.

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BT - The Netherlands as an EU Member

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A2 - Rood, J.

PB - Eleven International Publishing

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van Witteloostuijn A, Hendriks C. The paradox of The Netherlands: Why a successful economy is struggling? In Schout A, Rood J, editors, The Netherlands as an EU Member: Awkward or Loyal Partner?. The Hague / Portland, OR: Eleven International Publishing. 2013. p. 213-230