论文标题
$β= 6 $ Tracy-Widom分布和第二个Calogero-Painlevé系统
On $β=6$ Tracy-Widom distribution and the second Calogero-Painlevé system
论文作者
论文摘要
K. tokasaki于2001年引入了Calogero-Painlevé系统，这是对经典Painlevé方程的自然概括，以通过Calogero类型相互作用耦合到几种Painlevé“颗粒”。 2014年，I. Rumanov发现了一个了不起的事实，即第二个Calogero-PainlevéII方程的特定情况描述了具有偶数beta值的一般β-元素的Tracy-Widom分布函数。最近，在2017年的M. Bertola，M。Cafasso和V. Rubtsov的工作中，证明所有Calogero-Painlevé系统都可以放松，因此他们的解决方案承认了Riemann-Hilbert的代表性。 This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical $β=1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding至$β= 6 $ tracy-widom分布函数。 这里省略了一些技术细节，将在下一版本的文本中介绍。
The Calogero-Painlevé systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlevé equations to the case of the several Painlevé "particles" coupled via the Calogero type interactions. In 2014, I. Rumanov discovered a remarkable fact that a particular case of the second Calogero-Painlevé II equation describes the Tracy-Widom distribution function for the general beta-ensembles with the even values of parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlevé systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous asymptotic analysis of the Calogero-Painlevé equations which in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical $β=1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlevé system with a special focus on the Calogero-Painlevé system corresponding to $β= 6$ Tracy-Widom distribution function. Some technical details were omitted here and will be presented in the next version of the text.
