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Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on ℤ2

  • Omer Angel
  • Daniel de la Riva
  • Christoforos Panagiotis
  • Yuliang Shi
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Abstract

We prove continuity of the phase transition for long-range Bernoulli percolation on ℤ2 with connection probabilities that decay at least exponentially fast. This extends earlier work by Duminil-Copin, Sidoravicius, and Tassion on slabs. Our proof establishes a finite-size criterion that allows us to run a renormalisation argument. As a corollary, we obtain supercritical sharpness, in the sense that for every β>βc, there exists a unique macroscopic cluster in large boxes with high probability.

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Cite

BibTeXRIS
BibTeX
@misc{angel2026continuity,
  title = {{Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on $\mathbbZ^2$}},
  author = {Omer Angel and Daniel de la Riva and Christoforos Panagiotis and Yuliang Shi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35233},
  eprint = {2610.35233},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35233},
}
RIS
TY  - UNPB
TI  - Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on ℤ^2
AU  - Omer Angel
AU  - Daniel de la Riva
AU  - Christoforos Panagiotis
AU  - Yuliang Shi
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35233
UR  - https://arxiv.org/abs/2610.35233
AB  - We prove continuity of the phase transition for long-range Bernoulli percolation on ℤ^2 with connection probabilities that decay at least exponentially fast. This extends earlier work by Duminil-Copin, Sidoravicius, and Tassion on slabs. Our proof establishes a finite-size criterion that allows us to run a renormalisation argument. As a corollary, we obtain supercritical sharpness, in the sense that for every β>β_c, there exists a unique macroscopic cluster in large boxes with high probability.
ER  -