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Proof of a conjecture of N. Konno for the 1D contact process

J. van den Berg·O. Häggström·J. Kahn
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Abstract

Consider the one-dimensional contact process. About ten years ago, N. Konno stated the conjecture that, for all positive integers $n,m$, the upper invariant measure has the following property: Conditioned on the event that $O$ is infected, the events $\{$All sites $-n,...,-1$ are healthy$\}$ and $\{$All sites $1,...,m$ are healthy$\}$ are negatively correlated. We prove (a stronger version of) this conjecture, and explain that in some sense it is a dual version of the planar case of one of our results in \citeBHK.

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J. van den Berg, O. Häggström, J. Kahn. 2006-08-09. Proof of a conjecture of N. Konno for the 1D contact process. https://doi.org/10.1214/074921706000000031

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