Random Characters under the $L$-Measure, I: Dirichlet Characters
Random Characters under the $L$-Measure, I: Dirichlet Characters
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$L$-Measure 下的随机字符，I：狄利克雷字符
DOI：
10.1093/imrn/rnx168
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发表时间：
2019
影响因子：
1
通讯作者：
Barhoumi-Andréani Y
中科院分区：
文献类型：
--
作者：
Barhoumi-Andréani Y
我们将 Dirichlet 特征集上的-测度定义为 Plancherel 测度的类似物，Plancherel 测度曾被视为对称群不可约特征的测度。我们比较了这两种测度，并研究了当基础群规模增长时特征评估分布的极限。事实证明，这些评估在定律上收敛于柯西分布的虚数指数，其方式与复杂布朗运动的重新缩放绕组相同。这与对称群的情况形成对比，其中重整化特征在重新缩放后收敛于高斯分布（凯洛夫中心极限定理）。
We define the-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group.We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalized characters converge in law to Gaussians after rescaling (Kerov central limit theorem).