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Proof of a conjecture of N. Konno for the 1D contact process. (English) Zbl 1125.82023

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 16-23 (2006).
Summary: 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 \(-n,\dots,-1\) are healthy\(\}\) and {All sites \(1,\dots,m\) 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 [J. van den Berg, O. Häggstrøm and J. Kahn, Random Struct. Algorithms 29, No. 4, 417–435 (2006; Zbl 1112.60087)].
For the entire collection see [Zbl 1113.60008].

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
92D30 Epidemiology

Citations:

Zbl 1112.60087
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