Critical and near-critical influence bounds for ferromagnetic Ising models

Abstract

For a ferromagnetic Ising model on a graph of maximum degree Δ≥3, we prove a bound of order √n on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by β and ε=((Δ−1)tanhβ−1)₊, the bound is C(√n+nε). The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window ε=O(√(log n/n)), with the polynomial degree depending on the window size.

Near-critical Ising interfaces in the thermal direction: Absolute continuity with respect to SLE₃
Yu Feng
arXiv23 Oct 2026PreprintarXiv licenceFull textMathematical physics ProbabilityarXiv
Region bounds, error margins and certified positivity in the numerical conformal bootstrap; Or the first mathematically rigorous bounds on the 3d Ising CFT critical exponents
Marten Reehorst
arXiv14 Oct 2026PreprintarXiv licenceFull textMathematical physics High-energy & particle physicsarXiv
Near-optimal bounds for the critical 2d SHF and the delta-Bose gas
Xue-Mei Li, Te-Chun Wang
arXiv2 Nov 2026PreprintarXiv licenceFull textAnalysis & PDEs Mathematical physicsarXiv
An Õ(n² )-Time Sampler for Zero-Field Ferromagnetic Ising Models
Weiming Feng, Heng Guo, Yiyao Zhang
arXiv23 Sep 2026PreprintarXiv licenceFull textProbabilityarXiv
Comparing the Lee-Yang gap with the mass gap in the sub-critical planar Ising model
Noam Borgnia, Jui-Hui Chung, Jacob Shapiro
arXiv21 Sep 2026PreprintCC BY 4.0Full textMathematical physics ProbabilityarXiv

Cite

BibTeX RIS

BibTeX

@misc{pei2026critical,
  title = {{Critical and near-critical influence bounds for ferromagnetic Ising models}},
  author = {Yan Ru Pei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20866},
  eprint = {2609.20866},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20866},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Critical and near-critical influence bounds for ferromagnetic Ising models
AU  - Yan Ru Pei
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.20866
UR  - https://arxiv.org/abs/2609.20866
AB  - For a ferromagnetic Ising model on a graph of maximum degree Δ≥3, we prove a bound of order √n on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by β and ε=((Δ−1)tanhβ−1)_+, the bound is C(√n+nε). The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window ε=O(√(log n/n)), with the polynomial degree depending on the window size.
ER  -

Abstract

Related

Cite