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Continuity of the critical value and a shape theorem for long-range percolation

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  • Bäumler, Johannes

Abstract

Consider supercritical long-range percolation on Zd where two vertices x,y∈Zd are connected with probability asymptotic to ∥x−y∥−s for some s > 2d. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within n steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions d ≥ 2, the critical value depends continuously on the precise specifications of the model.

Suggested Citation

  • Bäumler, Johannes, 2026. "Continuity of the critical value and a shape theorem for long-range percolation," Stochastic Processes and their Applications, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:spapps:v:199:y:2026:i:c:s0304414926001146
    DOI: 10.1016/j.spa.2026.104982
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