Sharpness and Locality for Percolation on Finite Transitive Graphs

Abstract Let $(G_{n}) = \left ((V_{n},E_{n})\right )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that $\lvert V_{n} \rvert \to \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> as $n \to \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> . We say that percolation on $G_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> has a sharp phase transition (as $n \to \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> ) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on $G_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on $G_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in $G_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> coincides with the critical point for the emergence of an infinite cluster in the Benjamini-Schramm limit of $(G_{n})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math> , when this limit exists.