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Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps

  • Jing Liu
  • Kainan Xiang
  • Lang Zou
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Abstract

We study Bernoulli bond percolation under weak covering maps for a class of weighted inhomogeneous product measures such that each edge e carries a positive weight w(e) and is open at intensity λ≥ 0 with probability 1−exp(−λw(e)). We prove a strict monotonicity of the weighted critical intensities λc in the covering compatibility setting under bounded degree, uniform ellipticity, tame fibre and disjoint tree-lifting. While for the weighted uniform connection thresholds λconn, we establish their strict monotonicity for non-injective weak covering maps under bounded degree, covering compatibility, uniform ellipticity, bounded-fibre and disjoint tree-lifting. Moreover, we verify a general-profile theorem on strict monotonicities of the two thresholds λc and λconn respectively under explicit compact-interval finite-energy, derivative bounds and aforementioned geometric conditions.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026strict,
  title = {{Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps}},
  author = {Jing Liu and Kainan Xiang and Lang Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22417},
  eprint = {2610.22417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22417},
}
RIS
TY  - UNPB
TI  - Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps
AU  - Jing Liu
AU  - Kainan Xiang
AU  - Lang Zou
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22417
UR  - https://arxiv.org/abs/2610.22417
AB  - We study Bernoulli bond percolation under weak covering maps for a class of weighted inhomogeneous product measures such that each edge e carries a positive weight w(e) and is open at intensity λ≥ 0 with probability 1−exp(−λw(e)). We prove a strict monotonicity of the weighted critical intensities λ_c in the covering compatibility setting under bounded degree, uniform ellipticity, tame fibre and disjoint tree-lifting. While for the weighted uniform connection thresholds λ_conn, we establish their strict monotonicity for non-injective weak covering maps under bounded degree, covering compatibility, uniform ellipticity, bounded-fibre and disjoint tree-lifting. Moreover, we verify a general-profile theorem on strict monotonicities of the two thresholds λ_c and λ_conn respectively under explicit compact-interval finite-energy, derivative bounds and aforementioned geometric conditions.
ER  -