Dualities for the Domany–Kinzel Model

We study the Domany–Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p 1,p 2)∈[0,1]2. When p 1=αβ and p 2=α(2β−β 2) with (α,β)∈[0,1]2, the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities α of a site being open and β of a bond being open. This paper treats dualities for the Domany–Kinzel model ξ t A and the DKdual η t A starting from A. We prove that $$({\text{i}}){\text{ }}E(x^{ \shortmid \xi _t^A \cap B \shortmid } ) = E(x^{ \shortmid \xi _t^B \cap A \shortmid } ){\text{ if }}x = 1 - (2p_1 - p_2 )/p_1^2 ,{\text{ }}({\text{ii}}){\text{ }}E(x^{ \shortmid \xi _t^A \cap B \shortmid } ) = E(x^{ \shortmid \xi _t^B \cap A \shortmid } ){\text{ if }}x = 1 - (2p_1 - p_2 )/p_1 ,{\text{ and }}({\text{iii}}){\text{ }}E(x^{ \shortmid \eta _t^A \cap B \shortmid } ) = E(x^{ \shortmid \eta _t^B \cap A \shortmid } ){\text{ if }}x = 1 - (2p_1 - p_2 )$$ , as long as one of A,B is finite and p 2≤p 1.