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

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

2006-08-09

arxiv.org/abs/math/0608216v1

IMS Lecture Notes--Monograph Series 2006, Vol. 48, 16-23

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000031 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/

Scientific paper

10.1214/074921706000000031

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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