Jack Hanson - Universität Hamburg

Title: Chemical distances and k-point functions in high-dimensional percolation

Abstract: In 1984, Aizenman and Newman conjectured that k-point functions in high-dimensional critical percolation should behave as "simple combinations of the two-point function" governed by tree diagrams resembling those of a phi^3 field theory. We prove this conjecture. We also establish an asymptotic distributional law for the intrinsic or "chemical" distance in large critical clusters.

Johannes Baeumler - UCLA

Title: Arrival-time estimations for random recursive trees

Abstract: We estimate the arrival time of vertices in a uniform random recursive tree from its unlabeled structure. Using centrality-based rankings, we derive tail bounds for the relative estimation error that are uniform in the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability that it is smaller than the true arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation probability decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results identify a tradeoff between upper- and lower-tail performance in arrival-time estimation. Joint work with Simon Briend and Joost Jorritsma