高亏格平移曲面的 Benjamini-Schramm 极限：研究公告
Benjamini-Schramm limits of high genus translation surfaces: research announcement
AI总结：
本文证明面积等于亏格的 MSV 随机平移曲面在亏格趋于无穷时 Benjamini-Schramm 收敛于新引入的 Poisson 平移平面，并给出局部几何统计量的界。
摘要：
我们证明，面积等于亏格的 Masur-Smillie-Veech（MSV）分布随机平移曲面序列，在亏格趋于无穷时发生 Benjamini-Schramm 收敛。这意味着，对任意固定半径 $r>0$，若 $X_g$ 是面积为 $g$、亏格为 $g$ 的 MSV 分布随机平移曲面，且 $o$ 是 $X_g$ 中均匀随机选取的点，则 $X_g$ 中 $o$ 的半径 $r$ 邻域作为带点测度度量空间，依分布收敛于 Poisson 平移平面中根的半径 $r$ 邻域；Poisson 平移平面是我们在此引入的随机带点曲面。在此过程中，我们还得到了平移曲面统计局部几何性质的界，例如随机点 $o$ 的单射半径至多为 $r$ 的概率，这些结果本身可能具有独立意义。
英文摘要：
We prove that the sequence of Masur-Smillie-Veech (MSV) distributed random translation surfaces, with area equal to genus, Benjamini-Schramm converges as genus tends to infinity. This means that for any fixed radius $r>0$, if $X_g$ is an MSV-distributed random translation surface with area $g$ and genus $g$, and $o$ is a uniformly random point in $X_g$, then the radius-$r$ neighborhood of $o$ in $X_g$, as a pointed measured metric space, converges in distribution to the radius $r$ neighborhood of the root in a Poisson translation plane, which is a random pointed surface we introduce here. Along the way, we obtain bounds on statistical local geometric properties of translation surfaces, such as the probability that the random point $o$ has injectivity radius at most $r$, which may be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
