Some conditional correlation inequalities for percolation and related processes

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsSome Conditional Correlation Inequalities for Percolation and Related Processes ABSTRACT Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events ``there is an open path from s to a" and ``there is an open path from s to b" are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997). In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t.We also present analogues of some of our results for (a) randomcluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997).

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

“…. , 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]. …”

Scaling of a random walk on a supercritical contact process

“…We will use the fact that, for any λ ∈ (λ c , ∞), ν λ stochastically dominates a non-trivial Bernoulli product measure µ λ . This follows from Liggett and Steif [15], Theorem 1.2, Durrett and Schonmann [9], Theorem 1, and van den Berg, Häggström and Kahn [6], Theorem 3.5. Since Z δ (x) is monotone in ξ 0 , it is therefore enough to prove the statement under P µ λ .…”

“…We assume that 5) and that 6) i.e., the jump rate is constant and equal to γ everywhere, while the drift to the right is larger on infected sites than on healthy sites. Observe that the assumption in (1.6) is made without loss of generality: since the contact process is invariant under reflection in the origin, −W has the same law as W with inverted jump rates.…”

“…Therefore X nk,(n+1)k is equal to a (fixed) function of 6) which are jointly i.i.d. in n (when k is fixed).…”

Refinements of stochastic domination

“…The concept of downwards FKG was made explicit in [12], and was further studied in [2], where it was proved that the upper invariant measure for the contact process is downwards FKG.…”