From sine kernel to Poisson statistics
From sine kernel to Poisson statistics
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DOI：
10.1214/ejp.v19-3742
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发表时间：
2014-07
影响因子：
1.4
通讯作者：
Romain Allez;Laure Dumaz
中科院分区：
文献类型：
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作者：
Romain Allez;Laure Dumaz
我们研究了 Valko 和 Virag 中引入的正弦 beta 过程，此时逆温度 beta 趋于 0 。该点过程已被证明是大量β系综中特征值点过程的标度极限，其定律以不同角速度下布朗卡罗素的绕数来表征。经过对这一系列耦合扩散过程的仔细分析，我们证明了正弦-β点过程弱收敛于实线上的泊松点过程。因此，正弦-β点过程在刚性时钟（或栅栏）过程（对应于 $\beta=\infty$ ）和泊松过程之间建立了平滑的交叉。
We study the Sine beta process introduced in Valko and Virag, when the inverse temperature beta tends to 0 . This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of beta -ensembles and its law is characterised in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine-beta point process converges weakly to a Poisson point process on the real line . Thus, the Sine-beta point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\beta=\infty$ ) and the Poisson process.