### Abstract

For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λ

_{c}(ℤ^{2}) for the uniqueness threshold on the 2-dimensional integer lattice ℤ^{2}. 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 < λ_{c}(T_{Δ})λ < λ_{c}(T_{Δ}) where T_{Δ}is the infinite, regular tree of degree Δ. His result established a certain decay of correlations property called strong spatial mixing (SSM) on ℤ^{2}by proving that SSM holds on its self-avoiding walk tree T_{saw}(ℤ^{2}), and as a consequence he obtained that λ_{c}(Z^{2}) ≥ λ_{c}(T_{4}) = 1.675. Restrepo et al. (2011) improved Weitz’s approach for the particular case of ℤ^{2}and obtained that λ_{c}(ℤ^{2}) > 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T_{saw}(ℤ^{2}) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λ_{c}(ℤ^{2}) > 2.48.Original language | English |
---|---|

Title of host publication | The Netherlands as an EU Member |

Subtitle of host publication | Awkward or Loyal Partner? |

Editors | A. Schout, J. Rood |

Place of Publication | The Hague / Portland, OR |

Publisher | Eleven International Publishing |

Pages | 213-230 |

Number of pages | 312 |

ISBN (Print) | 9789490947996 |

Publication status | Published - 2013 |

## Fingerprint Dive into the research topics of 'The paradox of The Netherlands: Why a successful economy is struggling?'. Together they form a unique fingerprint.

## 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). Eleven International Publishing.