×

Some conditional correlation inequalities for percolation and related processes. (English) Zbl 1112.60087

The paper consists of three parts. Section 1 treats ordinary bond percolation on a finite or locally finite countably infinite graph \(G = (V,T)\) where each edge \(e \in E\), independently of the rest, is open with probability \(p_e\). The open cluster \(C_s\) consists of all edges in open paths starting at a vertex \(s \in V\). By Theorem 1.5, for vertices \(s\neq t\), conditional on the event \(\{s \nleftrightarrow t\}\) (nonexistence of open paths from \(s\) to \(t\)), the collection \(\{\mathbb{I}_{\{e \in C_s\}}: e \in E\}\cup \{1 - \mathbb{I}_{\{e \in C_t\}}: e \in E\}\) is positively associated. This covers results on conditional positive correlation by J. van den Berg and J. Kahn [Ann. Probab. 29, No. 1, 123–126 (2001; Zbl 1013.60079)]. Section 2, using different Markov chain techniques, gives extensions to random-cluster model. Section 3 deals with extensions to percolation on mixed graphs (allowing both directed and undirected edges) and contact processes. This improves some results of V. Belitsky, P. A. Ferrari, N. Konno and T. M. Liggett [Stochastic Processes Appl. 67, No. 2, 213–225 (1997; Zbl 0890.60094)]. There is interplay with the recent work of T. M. Liggett and J. E. Steif [Ann. Inst. Henri Poincaré, Probab. Stat. 42, No. 2, 223–243 (2006; Zbl 1087.60074)], involving so called downward FKG measures.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Ahlswede, Z Wahrsch Verw Geb 43 pp 183– (1978)
[2] Belitsky, Stoch Proc Appl 67 pp 213– (1997)
[3] van den Berg, Ann Probab 29 pp 123– (2001)
[4] , and, Proof of a conjecture of N. Konno for the 1D contact process,preprint, 2005.
[5] Combinatorics, Cambridge University Press, Cambridge, 1986. · Zbl 0595.05002
[6] Chayes, Stoch Proc Appl 65 pp 209– (1996)
[7] Percolation, 2nd ed., Springer-Verlag, 1999. · Zbl 0926.60004 · doi:10.1007/978-3-662-03981-6
[8] Percolative problems,in Probability andPhase Transition, (editor), Kluwer, Dordrecht, 1994, pp. 69–86. · Zbl 0830.60094 · doi:10.1007/978-94-015-8326-8_5
[9] The random-cluster model, in Probability onDiscrete Structures, (editor),Encyclopedia ofMathematical Sciences, Vol. 110, Springer-Verlag, 2003, pp. 73–123. · doi:10.1007/978-3-662-09444-0_2
[10] Häggström, Ann Appl Probab 9 pp 1149– (1999)
[11] Finite Markov Chains andAlgorithmic Applications, Cambridge University Press, Cambridge, 2002. · Zbl 0999.60001 · doi:10.1017/CBO9780511613586
[12] Harris, Ann Probab 2 pp 969– (1974)
[13] Holley, Comm Math Phys 36 pp 227– (1974)
[14] Phase Transitions of Interacting Particle Systems, World Scientific, Singapore, 1994.
[15] Interacting Particle Systems, Springer-Verlag, 1985. · Zbl 0559.60078 · doi:10.1007/978-1-4613-8542-4
[16] Stochastic interacting systems: Contact, Voter andExclusion Processes, Springer-Verlag, 1999. · doi:10.1007/978-3-662-03990-8
[17] Survival and coexistence in interacting particle systems,Probability and Phase Transition, Kluwer, Dordrecht, 1994, pp. 209–226. · Zbl 0832.60094
[18] and, Stochastic Domination: The Contact Process, Ising Models and FKG Measures,Annales Institut H. Poincaré, Probabilités et Statistiques,to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.