All articles tagged with #transitive graphs
A five-person ETH Zurich team proved supercritical sharpness for percolation on all infinite transitive graphs, showing that any probability just above the critical threshold yields almost full connectivity and solving a decades-long puzzle in probability and graph theory, using a refined sprinkling approach that unifies the result across graph types.
Mathematicians have made significant progress in proving Schramm's locality conjecture, which relates to the percolation threshold in transitive graphs. The conjecture states that the percolation threshold can be determined solely by the close-up perspective of the graph. Two groups of mathematicians have successfully tackled the conjecture for fast-growth and slow-growth graphs, but the challenge lies in addressing graphs with intermediate growth rates. Easo and Hutchcroft have extended their results to cover these graphs, completing the trichotomy and proving the conjecture. This breakthrough provides insights into the behavior above and below the percolation threshold, but the exact behavior at the threshold remains an open question for most graphs.