Electronic Journal of Probability

Disagreement percolation for the hard-sphere model

Hofer-Temmel Christoph

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Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 91, 22 pp.

Received: 19 March 2018
Accepted: 12 May 2019
First available in Project Euclid: 10 September 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B21: Continuum models (systems of particles, etc.)
Secondary: 60E15: Inequalities; stochastic orderings 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes 82B43: Percolation [See also 60K35] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

hard-sphere model disagreement percolation unique Gibbs measure stochastic domination Boolean model absence of phase transition dependent thinning

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Christoph, Hofer-Temmel. Disagreement percolation for the hard-sphere model. Electron. J. Probab. 24 (2019), paper no. 91, 22 pp. doi:10.1214/19-EJP320. https://projecteuclid.org/euclid.ejp/1568080871

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