AI 中文总结
该研究证明了所有β>0时正弦贝塔点过程是尾平凡的，利用DLR方程、相关密度等推导得出，进而得到其在吉布斯测度类中的极值性、扩散的L²遍历性及碰撞二分法等应用。
AI 中文摘要
我们证明，对于所有β>0，正弦贝塔点过程是尾平凡的。证明的关键要素是某个一致熵界，该界利用了Dereudre等人（2021）的DLR方程，以及Qu和Valko（2025）、Assiotis和Najnudel（2026）得到的有界两点相关密度，这使我们能够证明尾事件是平移不变的。尾平凡性随后可由Assiotis（2026）的唯一性定理，或由我们从Dumaz和Malvy（2026）的相关估计中推导得到的平移混合性得出。作为推论，我们得到以下应用：(a) 正弦贝塔在无平稳性和有限能量条件的固定核吉布斯测度的全凸类中的极值性；(b) 相关无限粒子可逆扩散对于所有β>0的L²遍历性；(c) 该扩散的几乎必然碰撞/非碰撞二分法，阈值为β=1。
英文摘要
We prove that the sine beta point process is tail trivial for every beta >0. The key ingredient in the proof is a certain uniform entropy bound, that makes use of the DLR equations of Dereudre et al. (2021) and the bounded two-point correlation density obtained by Qu and Valko (2025) and Assiotis and Najnudel (2026), that allows us to show that tail events are translation invariant. Tail triviality then follows either from the uniqueness theorem of Assiotis (2026) or from translation mixing, which we derive from the correlation estimates of Dumaz and Malvy (2026). As consequences, we obtain the following applications: (a) extremality of sine beta in the full convex class of fixed kernel Gibbs measures without stationarity and finite-energy conditions; (b) L^2-ergodicity of the associated infinite-particle reversible diffusion for every beta>0; and (c) the almost-sure collision/noncollision dichotomy with threshold beta=1 for the diffusion.
