## Electronic Journal of Probability

### Continuum percolation for Gibbsian point processes with attractive interactions

Sabine Jansen

#### Abstract

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large $\beta$). The main results are bounds on percolation thresholds $\rho _\pm (\beta )$ in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster size distributions. This formula establishes a link with the Gibbs variational principle and a form of equivalence of ensembles, and allows us to combine knowledge on finite volume, canonical Gibbs measures with infinite volume, grand-canonical Gibbs measures

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 47, 22 pp.

Dates
Accepted: 25 February 2016
First available in Project Euclid: 28 July 2016

https://projecteuclid.org/euclid.ejp/1469720442

Digital Object Identifier
doi:10.1214/16-EJP4175

Mathematical Reviews number (MathSciNet)
MR3539641

Zentralblatt MATH identifier
1385.60059

#### Citation

Jansen, Sabine. Continuum percolation for Gibbsian point processes with attractive interactions. Electron. J. Probab. 21 (2016), paper no. 47, 22 pp. doi:10.1214/16-EJP4175. https://projecteuclid.org/euclid.ejp/1469720442

#### References

• [1] M. Aizenman, J. Bricmont, and J.L. Lebowitz, Percolation of the minority spins in high-dimensional Ising models, J. Stat. Phys. 49 (1987), 859–865.
• [2] D. Aristoff, Percolation of hard disks, J. Appl. Probab. 51 (2014), 235–246.
• [3] X. Blanc, Lower bound for the interatomic distance in Lennard-Jones clusters, Comput. Optim. Appl. 29 (2004), 5–12.
• [4] D. Brydges and P. Federbush, A new form of the Mayer expansion in classical statistical mechanics, J. Math. Phys. 19 (1978), 2064–2067.
• [5] D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. II, second ed., Probability and its Applications (New York), Springer, New York, 2008.
• [6] B.N.B. de Lima and A. Procacci, The Mayer series of the Lennard-Jones gas: improved bounds for the convergence radius, J. Stat. Phys. 157 (2014), 422–435.
• [7] A. Dembo and O. Zeitouni, Large deviations techniques and applications, second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998.
• [8] R. L. Dobrushin, Gibbsian random fields for particles without hard core, Teoret. Mat. Fiz. 4 (1970), 101–118.
• [9] H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction, Probab. Theory Relat. Fields 99 (1994), 171–195.
• [10] H. The equivalence of ensembles for classical particle systems, J. Statist. Phys. 80 (1995), 1341–1378.
• [11] H.-O. Georgii, O. Häggström, and C. Maes, The random geometry of equilibrium phases, Phase transitions and critical phenomena, Vol. 18, Phase Transit. Crit. Phenom., vol. 18, Academic Press, San Diego, CA, 2001, pp. 1–142.
• [12] H.-O. Georgii and H. Zessin, Large deviations and the maximum entropy principle for marked point random fields, Probab. Theory Related Fields 96 (1993), 177–204.
• [13] S. Jansen, Mayer and virial series at low temperature, J. Stat. Phys. 147 (2012), 678–706.
• [14] S. Jansen, W. König, and B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, Ann. Appl. Probab. 25 (2015), 930–973.
• [15] R. Meester and R. Roy, Continuum percolation, Cambridge Tracts in Mathematics, vol. 119, Cambridge University Press, Cambridge, 1996.
• [16] T. Morais, A. Procacci, and B. Scoppola, On Lennard-Jones type potentials and hard-core potentials with an attractive tail, J. Stat. Phys. 157 (2014), 17–39.
• [17] M. G. Mürmann, Equilibrium distributions of physical clusters, Comm. Math. Phys. 45 (1975), 233–246.
• [18] E. Pechersky and A. Yambartsev, Percolation properties of the non-ideal gas, J. Stat. Phys. 137 (2009), no. 3, 501–520.
• [19] S. Poghosyan and D. Ueltschi, Abstract cluster expansion with applications to statistical mechanical systems, J. Math. Phys. 50 (2009), 053509, 17.
• [20] D. Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
• [21] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970), 127–159.
• [22] K. Stucki, Continuum percolation for Gibbs point processes, Electron. Commun. Probab. 18 (2013), no. 67, 10.
• [23] F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys. 262 (2006), 209–236.
• [24] H. Zessin, A theorem of Michael Mürmann revisited, Izv. Nats. Akad. Nauk Armenii Mat. 43 (2008), 69–80, translation in J. Contemp. Math. Anal. 43 (2008), no. 1, 50–58.