The non-uniform dispersion process on the infinite integer line is a synchronous process where n particles are placed at the origin initially, and any particle not exclusively occupying an integer site will move at the next time step to the right adjacent integer with probability $p_n$ and to the left with probability $1-p_n$ independently. We characterize the longest distance from the origin when the dispersion process stops, which is shown to be $\mathrm{\Theta }(n)$ with high probability for fairly general $p_n$.

中文翻译：

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无限线上不均匀分散过程的最长距离

无限整数线上的非均匀分散过程是一个同步过程，其中最初将n个粒子放置在原点，并且任何不排他地占据整数位置的粒子将在下一个时间步移动到右邻整数$p_{ñ}$ 并向左概率 $\mathrm{1个}-p_{ñ}$独立地。当分散过程停止时，我们描述了距原点的最长距离，这表明$\mathrm{\Theta }（ñ）$ 相当普遍的可能性很高 $p_{ñ}$。