Unveiling the Infinite Graph's 'Melting' Point Up Close

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.