Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $\mathbf{Z}_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Fig. 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j) \in \mathbf{Z}_+^2 $ at a given time $k\ge 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\rightarrow \infty $ . The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $\mathbf{Z}$ , four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of his neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.