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

van den Berg, J., CWI and VUA; Häggström, O., Chalmers University of Technology; Kahn, J., Rutgers University

doi:10.1214/074921706000000031

VOL. 48 | 2006 Proof of a conjecture of N. Konno for the 1D contact process

J. van den Berg, O. Häggström, J. Kahn

Editor(s) Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy

IMS Lecture Notes Monogr. Ser., 2006: 16-23 (2006) DOI: 10.1214/074921706000000031

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 $-n, \ldots, -1$ are healthy$\}$ and $\{$All sites $1, \ldots, m$ 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 van den Berg, J., Häggström, O., Kahn, J. (2005), Some conditional correlation inequalities for percolation and related processes, to appear in Random Structures and Algorithms.