同调类中模子格的闭测地线
Closed geodesics in homology classes modulo sublattices
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中文总结 AI 辅助
研究Weil-Petersson随机双曲曲面中同调类模素指数格的本原闭测地线分布，计算加权计数函数中心矩、分析方差渐近行为，结果类同算术级数素数分布现象。
中文摘要 AI 辅助
设M为亏格g的Weil-Petersson随机双曲曲面，Γ⊂Z^(2g)是素指数q的格。我们研究大亏格极限下，同调类中模Γ的本原闭测地线的分布。对所有指数为q的格取平均（q→∞），计算对应加权计数函数的所有中心矩，并展示泊松与高斯区间之间的转变（取决于X/(q log X)，即给定同调类中模Γ的本原测地线的期望数，是否趋向λ>0或∞）。我们还研究同调类计数的未归一化方差G_M(X,Γ)，并证明当X→∞时，在大亏格极限下对所有素指数q的格取平均，该方差渐近于X log X。这些结果类似于算术级数中素数分布的相关现象。
英文摘要
Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $Γ\subset \mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $Γ$ in the large genus limit. Averaging over all lattices of index $q$, with $q \to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $Γ$, tends to $λ>0$ or $\infty$). We also study the unnormalized variance $G_M(X,Γ)$ of the counts among homology classes, and show that as $X \to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.
