# On Maximal Hard-Core Thinnings of Stationary Particle Processes

- 77 Downloads

## Abstract

The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally unique. More precisely, distributional uniqueness holds in subcritical and barely supercritical regimes of continuum percolation. Additionally, based on the analysis of a specific example, we argue that fluctuations in grain sizes can play an important role for establishing distributional uniqueness at high intensities. Finally, we provide a family of algorithmically constructible approximations whose volume fractions are arbitrarily close to the maximum.

## Keywords

Particle process Hard-core process Stochastic domination Thinning## Mathematics Subject Classification

60G55 60D05## Notes

### Acknowledgements

We thank the anonymous referee for the time and energy invested into producing a very thorough and detailed report. The suggestions and comments made in this report substantially improved the quality of the article. This research publication was funded by LMU Munich’s Institutional Strategy LMUexcellent within the framework of the German Excellence Initiative.

## References

- 1.Aizenman, M., Grimmett, G.R.: Strict monotonicity for critical points in percolation and ferromagnetic models. J. Stat. Phys.
**63**(5–6), 817–835 (1991)ADSMathSciNetCrossRefGoogle Scholar - 2.Daley, D.J., Vere-Jones, D.D.: An Introduction to the Theory of Point Processes I/II. Springer, New York (2005/2008)Google Scholar
- 3.Franceschetti, M., Penrose, M.D., Rosoman, T.: Strict inequalities of critical values in continuum percolation. J. Stat. Phys.
**142**(3), 460–486 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 4.Gamarnik, D., Nowicki, T., Swirszcz, G.: Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Random Struct. Algorithms
**28**(1), 76–106 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Gentner, D., Last, G.: Palm pairs and the general mass-transport principle. Math. Z.
**267**(3), 695–716 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Grimmett, G.R.: Percolation, 2nd edn. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
- 7.Grimmett, G.R., Stacey, A.M.: Critical probabilities for site and bond percolation models. Ann. Probab.
**26**(4), 1788–1812 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Hall, P.: Introduction to the Theory of Coverage Processes. Wiley, New York (1988)zbMATHGoogle Scholar
- 9.Heydenreich, M., Merkl, F., Rolles, S.W.W.: Spontaneous breaking of rotational symmetry in the presence of defects. Electron. J. Probab.
**19**, 1–17 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Hofer-Temmel, C., Houdebert, P.: Disagreement percolation for marked Gibbs point processes arXiv preprint arXiv:1709.04286 (2017)
- 11.Holroyd, A.E., Peres, Y.: Trees and matchings from point processes. Electron. Commun. Probab.
**8**, 17–27 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Hörig, M.: Zufällige harte Partikelsysteme. PhD thesis, KIT (2010)Google Scholar
- 13.Hörig, M., Redenbach, C.: The maximum volume hard subset model for Poisson processes: simulation aspects. J. Stat. Comput. Simul.
**82**(1), 107–121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Hug, D., Schneider, R.: Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal.
**17**(1), 156–191 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)Google Scholar
- 16.Kallenberg, O.: Random Measures, Theory and Applications. Springer, Cham (2017)Google Scholar
- 17.Last, G., Penrose, M.D.: Lectures on the Poisson Process. Cambridge University Press (2017)Google Scholar
- 18.Last, G., Thorisson, H.: Invariant transports of stationary random measures and mass-stationarity. Ann. Probab.
**37**(2), 790–813 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab.
**25**(1), 71–95 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Mase, S., Møller, J., Stoyan, D., Waagepetersen, R.P., Döge, G.: Packing, densities and simulated tempering for hard core Gibbs point processes. Ann. Inst. Stat. Math.
**53**(4), 661–680 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Menshikov, M.V.: Quantitative estimates and strong inequalities for the critical points of a graph and its subgraph. Teor. Veroyatnost. i Primenen.
**32**(3), 599–602 (1987)MathSciNetGoogle Scholar - 22.Merkl, F., Rolles, S.W.W.: Spontaneous breaking of continuous rotational symmetry in two dimensions. Electron. J. Probab.
**14**, 1705–1726 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab.
**11**(4), 1005–1041 (2001)MathSciNetzbMATHGoogle Scholar - 24.Penrose, M.D., Yukich, J.E.: Limit theory for random sequential packing and deposition. Ann. Appl. Probab.
**12**(1), 272–301 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
- 26.Timár, Á.: Tree and grid factors for general point processes. Electron. Commun. Probab.
**9**(9), 53–59 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Torquato, S.: Random Heterogeneous Materials. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
- 28.van den Berg, J., Steif, J.E.: Percolation and the hard-core lattice gas model. Stoch. Process. Appl.
**49**(2), 179–197 (1994)MathSciNetCrossRefzbMATHGoogle Scholar