We study quasi-isometric embeddings from a random Bernoulli percolation sample on the product of two regular trees into the product itself, and show some rigidity properties that can be seen as an extension of quasi-isometric rigidity of higher rank non-uniform lattices. We also prove that two independent samples are almost surely not quasi-isometric equivalent, thus confirming that such a phenomenon occurs in the higher-rank setting, as conjectured by Abert. Joint work with Zhiqiang Li and Ranfeng Yu.
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