随机哈密顿量 II：中心极限定理与随机游走的霍弗几何
Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks
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中文总结 AI 辅助
本文利用随机游走研究哈密顿微分同胚群的整体几何，证明其随机游走期望霍弗范数增长与步数平方根相关，给出相关测度、可测性及中心极限定理的结果。
中文摘要 AI 辅助
本文利用随机游走研究哈密顿微分同胚群$\text{Ham}(M,\boldsymbol{\u03c9})$的整体几何。在一大类辛流形上，我们证明此类随机游走的期望霍弗范数增长速度至少与步数的平方根成正比。此外，我们证明若随机游走限制在阿贝尔子群中，其增长速度也被步数的平方根从上界约束。我们还提供了一些数值证据，表明该上界在非交换子群处不成立。这为哈密顿微分同胚群的子群提供了有限维李群中交换性平坦性的概率版本。在研究过程中，我们证明了前序论文中引入的概率测度类是$\text{Ham}(M,\boldsymbol{\u03c9})$上$C^\text{\u221e}$拓扑下的博雷尔测度类，证明了稳定交换子长度的可测性，并证明了$\text{Ham}(M,\boldsymbol{\u03c9})$上霍弗-利普希茨拟同态的中心极限定理。
英文摘要
This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,ω)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,ω)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,ω)$.
发表机构
- University of Cambridge（剑桥大学）
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