AI 中文总结
该研究证明了一维Bak-Sneppen模型中两种更新规则下雪崩相变的尖锐性，确定了雪崩临界阈值并验证了猜想的平稳乘积极限。
AI 中文摘要
我们证明了一维Bak-Sneppen模型中雪崩相变的尖锐性，分别针对单侧更新规则和各向同性更新规则。在单侧模型中，雪崩范围由非线性自相互作用跳跃过程描述，其尾部满足三角系统。针对递减概率质量的极值不等式给出了 susceptibility 阈值处范围尾部的多项式下界；该下界迫使无限雪崩在临界点以上立即出现，从而确定了 susceptibility 和无限雪崩阈值。通过雪崩端点的比较，该等式可推广到各向同性模型。在两种模型中，临界点处均无无限雪崩发生，而在每个严格亚临界水平上，范围具有指数矩。结合Meester和Znamenski的范围-持续时间及平稳极限结果，这确定了所有三个雪崩临界阈值，并证明了两种模型中猜想的平稳乘积极限。
英文摘要
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.
