For percolation on T times Z^d and on lamplighter graphs over a tree, the Hausdorff dimension of cluster limit sets jumps from at most 1/2 to 1 at the uniqueness threshold, and four critical thresholds coincide.
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Dimension jump at the uniqueness threshold for percolation in $\infty+d$ dimensions
For percolation on T times Z^d and on lamplighter graphs over a tree, the Hausdorff dimension of cluster limit sets jumps from at most 1/2 to 1 at the uniqueness threshold, and four critical thresholds coincide.