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标题: "Sharpness of the avalanche phase transition in the Bak--Sneppen model"
原文链接: "https://arxiv.org/abs/2608.10992"
发布日期: "2026-08-11"
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---

## Mathematics > Probability

## Title:Sharpness of the avalanche phase transition in the Bak--Sneppen model

Authors:[Olivier Hénard](https://arxiv.org/search/math?searchtype=author&query=H%C3%A9nard,+O)

[View PDF](https://arxiv.org/pdf/2608.10992) [HTML (experimental)](https://arxiv.org/html/2608.10992v1)

> Abstract:We prove sharpness of the avalanche phase transition in the one-dimensional Bak--Sneppen model, for both the one-sided and isotropic update rules. In the one-sided model, the avalanche range is described by a nonlinear self-interacting jump process whose tails satisfy a triangular system. An extremal inequality for decreasing probability masses yields a polynomial lower bound on the range tail at the susceptibility threshold; this bound forces the appearance of infinite avalanches immediately above criticality and hence identifies the susceptibility and infinite-avalanche thresholds. A comparison of avalanche endpoints transfers this equality to the isotropic model. In both models, no infinite avalanche occurs at criticality, while the range has exponential moments at every strictly subcritical level. Combined with the range--duration and stationary-limit results of Meester and Znamenski, this identifies all three avalanche critical thresholds and proves the conjectured stationary product limit in both models.

| Subjects: | Probability (math.PR) |
| --- | --- |
| Cite as: | [arXiv:2608.10992](https://arxiv.org/abs/2608.10992) \[math.PR\] |
|  | (or [arXiv:2608.10992v1](https://arxiv.org/abs/2608.10992v1) \[math.PR\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.10992](https://doi.org/10.48550/arXiv.2608.10992) |

## Submission history

From: Olivier Hénard \[[view email](https://arxiv.org/show-email/d255d84e/2608.10992)\]  
**\[v1\]** Tue, 11 Aug 2026 14:45:25 UTC (27 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.10992) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
