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A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed atthe axes, starting from an arbitrary point $(i_0,j_0)$ and with stepprobabilities drawn on Figure 1 is considered. The trivariate generatingfunction of probabilities that the random walk hits a given point $(i,j)\inZ_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorptionat a given time $k$ and at a given axis are found, and their precise asymptoticis derived as the time $k\to\infty$. The equivalence of two typical ways ofconditioning this random walk to never reach the axes is established. Theresults are also applied to the analysis of the voter model with two candidatesand initially, in the population $Z$, four connected blocks of same opinions.Then, a citizen changes his mind at a rate proportional to the number of itsneighbors that disagree with him. Namely, the passage from four to two blocksof opinions is studied.

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