Chemical distance in hierarchical percolation

ESB 4127

Hierarchical percolation is a toy model for percolation on $\mathbb{Z}^d$ that, much like the Euclidean model, is expected to exhibit mean-field behavior in high dimensions, non-mean-field behavior in low dimensions, and mean-field behavior with logarithmic corrections at the upper-critical dimension. Building on Hutchcroft’s work on cluster volumes in hierarchical percolation, we examine the distribution of the chemical distance in high dimensions and the upper-critical dimension. In this talk, I will explain the renormalization group—style analysis we use to obtain precise estimates on the moments of the chemical distance and discuss our attempt to generalize this approach to long-range percolation on the Euclidean lattice. Joint work with Tom Hutchcroft.