Hierarchical percolation is a toy model for percolation on Z^d that, much like percolation on Euclidean lattices, is expected to exhibit mean-field behavior in high dimensions, non-mean-field behavior in low dimensions, and logarithmic corrections to mean-field behavior at the upper-critical dimension. The hierarchical lattice allows for a renormalization group—style analysis which is currently inaccessible for percolation on Euclidean lattices. Building on Hutchcroft’s work on cluster volumes in all dimensions, we examine the distribution of the chemical distance, extremal distance (also known as the effective resistance), and pivotal distance in high dimensions and the upper-critical dimension. Joint work with Tom Hutchcroft.
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