We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over ${\Bbb Z}^d$ ($d \ge 3$) is the Poisson boundary. For $d \ge 5$, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

中文翻译：

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点灯者群的泊松边界：Kaimanovich-Vershik 猜想的证明

我们正面回答了 Kaimanovich 和 Vershik 在 1979 年提出的问题，表明在 ${\Bbb Z}^d$ ($d \ge 3$) 上的点灯组上进行简单随机行走的灯的最终配置是泊松边界。对于 $d \ge 5$，Erschler (2011) 早先已经证明了这一点。我们将其扩展到更一般组的更一般类型的步行。