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

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 [2].

Extreme paths in oriented two-dimensional percolation

“…In this section we give another proof of Theorem 1.1, based on a Markov Chain introduced in [2]. Assume A ⊂ L 0 , B ⊂ L n and E is a set of oriented edges in Λ containing at least one path from A to B.…”

“…In Section 2 we provide a proof of our main theorem, based on a Markov chain that was introduced in [5]. In that paper, the Markov chain was used to prove that the one-dimensional nearest-neighbor contact process satisfies the following property.…”

“…In section we provide a proof of Theorem 1, based on a Markov chain introduced in [5]. Without loss of generality, we assume that m = 0 and we let A and B be finite subsets of L 0 and L n , respectively, such that (A, B) = ∅.…”

“…In the sequel we will need to consider this Markov chain for different sets E. We will call it the Markov chain associated to E. Besides this, we extend the notation of the previous sections by letting E (A, B) be the set of paths from A to B whose edges are in E. Proposition 1. (Van den Berg et al [5].) The measure P p (• | T ) is invariant for the Markov chain.…”

Problem Solving is Often a Matter of Cooking Up an Appropriate Markov Chain*

“…(2006a), theorem 2 is proved in the greater generality of the random‐cluster model with clustering parameter q 1 (the case q =1 corresponds to the ordinary bond percolation setup considered here) using an extension of the Markov chain argument; the induction‐based alternative proof seems not to work in this setting. Applications of theorem 2 to settle certain open problems concerning the equilibrium behaviour of an interacting particle system known as the contact process appear in van den Berg et al. (2006a, b).…”