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.
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机译：考虑从内部任意位置$（i_0，j_0）$开始并在图1上绘制阶跃概率的，内部空间均匀，在轴上吸收的$ Z _ + ^ 2 $的随机游动。明确了随机游走在给定时间$ k \ geq 0 $到达给定点$（i，j）\ inZ _ + ^ 2 $的概率的三变量生成函数。找到在给定时间$ k $和在给定轴上的吸收概率，并将其精确渐近线导出为时间$ k \ to \ infty $。建立了两种典型的调节这种随机游动以永不到达轴的方式的等效性。结果也被用于分析具有两个候选人的选民模型，最初是在人口$ Z $中，四个具有相同意见的关联区块，然后，一个公民改变了主意，其比例与拒绝他的邻居数量成正比。即，研究了从四个观点到两个观点的转变。
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