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The critical percolation probability is local

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arxiv 2310.10983 v1 pith:KMULIPBP submitted 2023-10-17 math.PR

Philip Easo , Tom Hutchcroft This is my paper · ORCID
classification math.PR
keywords vertex-transitivegraphsinfinitecriticaleverygraphone-dimensionalpercolation
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abstract

We prove Schramm's locality conjecture for Bernoulli bond percolation on transitive graphs: If $(G_n)_{n\geq 1}$ is a sequence of infinite vertex-transitive graphs converging locally to a vertex-transitive graph $G$ and $p_c(G_n) \neq 1$ for every $n \geq 1$ then $\lim_{n\to\infty} p_c(G_n)=p_c(G)$. Equivalently, the critical probability $p_c$ defines a continuous function on the space $\mathcal{G}^*$ of infinite vertex-transitive graphs that are not one-dimensional. As a corollary of the proof, we obtain a new proof that $p_c(G)<1$ for every infinite vertex-transitive graph that is not one-dimensional.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Local limit of Prim's algorithm

    math.PR 2025-07 conditional novelty 8.0 of 10

    Running Prim's algorithm for tn+o(n) steps on a locally convergent weighted graph sequence converges in local process convergence to the expanded invasion percolation cluster of the limit graph.

  2. From local giants to locality in long-range percolation

    math.PR 2026-07 conditional novelty 7.0 of 10

    Long-range percolation on polynomial-growth transitive graphs is local for α∈(0,2), and a new Voronoi-tile renormalization yields giant-cluster, cluster-decay, isoperimetric and transience results.

  3. Concept-wise Attention for Fine-grained Concept Bottleneck Models

    cs.CV 2026-04 unverdicted novelty 7.0 of 10

    CoAt-CBM uses concept-wise visual queries and concept contrastive optimization to improve fine-grained image–concept alignment in CLIP-based concept bottleneck models.

  4. Percolation on random 2-lifts

    math.PR 2025-06 conditional novelty 6.0 of 10

    For random 2-lifts of transitive graphs, the percolation critical point is continuous in the switching probability q, strictly smaller than the base threshold, and subcritical clusters decay exponentially at q=1/2.

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