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PreprintCC BY 4.0HAL

High-dimensional critical percolation and critical Branching random walks are intersection equivalent

  • Bruno Schapira
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Abstract

We prove that on Z^d , with d> 6, and under the usual assumption that the two-point function and the Green's function are of the same order, the probabilities to hit any finite set for a critical Branching random walk starting from the origin and the critical percolation cluster of the origin are within a positive multiplicative constant one from each other, as long as the set is sufficiently far away from the origin comparatively to its diameter. The proof relies on the introduction of a truncated equilibrium measure in percolation and the derivation of an approximate last passage formula.

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Cite

BibTeXRIS
BibTeX
@misc{schapira2026highdimensional,
  title = {{High-dimensional critical percolation and critical Branching random walks are intersection equivalent}},
  author = {Bruno Schapira},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37133},
  eprint = {2610.37133},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05838269v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - High-dimensional critical percolation and critical Branching random walks are intersection equivalent
AU  - Bruno Schapira
PY  - 2026
DA  - 2026/10/25
PB  - HAL
DO  - 10.48550/arxiv.2610.37133
UR  - https://hal.science/hal-05838269v1
AB  - We prove that on Z^d , with d> 6, and under the usual assumption that the two-point function and the Green's function are of the same order, the probabilities to hit any finite set for a critical Branching random walk starting from the origin and the critical percolation cluster of the origin are within a positive multiplicative constant one from each other, as long as the set is sufficiently far away from the origin comparatively to its diameter. The proof relies on the introduction of a truncated equilibrium measure in percolation and the derivation of an approximate last passage formula.
ER  -