×

zbMATH — the first resource for mathematics

A strong correlation inequality for contact processes and oriented percolation. (English) Zbl 0890.60094
The strengthening of the positive correlation inequality \(\nu (A\cap B)\nu (A\cup B) \geq \nu (A)\nu (B)\) is proved for the following two cases. The extinction probability \(\nu (A)\) for the contact process on a countable set \(S\) with initial state \(A\subset S\), or equivalently, for \(\nu (A)=\nu _\infty (\{\eta ;\eta _A\equiv 0\})\) with \(\nu _\infty \) being the upper invariant measure of the contact process. The same inequality is independently proved for \(\nu \) being the extinction probability of an oriented percolation which can be viewed as a discrete time version of the one-dimensional contact process.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Durrett, R., Oriented percolation in two dimensions, Ann. probab., 12, 999-1040, (1984) · Zbl 0567.60095
[2] Durrett, R., Lecture notes on particle systems and percolation, (1988), Wadsworth & Brooks Cole Belmont, CA · Zbl 0659.60129
[3] Harris, T.E., Contact interactions on a lattice, Ann. probab., 2, 969-988, (1974) · Zbl 0334.60052
[4] Harris, T.E., A correlation inequality for Markov processes in partially ordered state spaces, Ann. probab., 5, 451-454, (1977) · Zbl 0381.60072
[5] Konno, N., Phase transitions of interacting particle systems, (1994), World Scientific Singapore · Zbl 0847.60088
[6] Liggett, T.M., Interacting particle systems, (1985), Springer Berlin · Zbl 0559.60078
[7] Liggett, T.M., The survival of one-dimensional contact processes in random environments, Ann. probab., 20, 696-723, (1992) · Zbl 0754.60126
[8] Liggett, T.M., Survival and coexistence in interacting particle systems, (), 209-226 · Zbl 0832.60094
[9] Walter, W., Differential and integral inequalities, (1970), Springer Berlin
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.