Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 46 min ago
PreprintCC BY 4.0HAL

On the number of pioneers in high-dimensional critical percolation

  • Amine Asselah
  • Bruno Schapira
Open at sourcePDFDOICite

Abstract

We prove that for critical percolation in high dimension the expected number of pioneers on any set containing the origin is of the same order as the probability that a simple random walk starting from the origin exits the set. Our proof works in all dimensions d > 6, under the usual assumption that the two-point function is of the same order as the Green's function. As an application we obtain a sharp uniform quantitative bound on the probability that the cluster of the origin intersects any given ball.

Related

  1. High-dimensional critical percolation and critical Branching random walks are intersection equivalent
    Bruno Schapira
    HALPreprintCC BY 4.0Full textProbabilityHALarXiv
  2. Super-Brownian limits and the k-point function for high-dimensional percolation
    Arthur Blanc-Renaudie, Tom Hutchcroft
    HALPreprintLicence not statedGeneral mathematicsHAL
  3. On the critical value of void percolation in high dimensions
    Patrick Charbonneau, Marcus Michelen
    arXivPreprintarXiv licenceFull textProbabilityCondensed matterarXiv
  4. A new approach for estimating the number of spikes in high-dimensional Gaussian models
    Hugo Jeannin, Pascal Vallet, Florent Bouchard +2
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  5. Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
    Shunqin Zhang, Jiqiang Zheng, Yunrui Zheng
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{asselah2026number,
  title = {{On the number of pioneers in high-dimensional critical percolation}},
  author = {Amine Asselah and Bruno Schapira},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35553},
  eprint = {2610.35553},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05839694v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the number of pioneers in high-dimensional critical percolation
AU  - Amine Asselah
AU  - Bruno Schapira
PY  - 2026
DA  - 2026/10/26
PB  - HAL
DO  - 10.48550/arxiv.2610.35553
UR  - https://hal.science/hal-05839694v1
AB  - We prove that for critical percolation in high dimension the expected number of pioneers on any set containing the origin is of the same order as the probability that a simple random walk starting from the origin exits the set. Our proof works in all dimensions d > 6, under the usual assumption that the two-point function is of the same order as the Green's function. As an application we obtain a sharp uniform quantitative bound on the probability that the cluster of the origin intersects any given ball.
ER  -