论文标题
限制较大维度和自由卷积的Bessel和Dunkl过程的定理
Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions
论文作者
论文摘要
我们在$ \ mathbb r^n $上研究贝塞尔和dunkl进程$（x_ {t，k}）_ {t \ ge0} $，其中可能是多变量耦合常数$ k \ ge0 $。这些过程描述了具有$ N $颗粒的Calogero-Moser-Sutherland类型的相互作用粒子系统。对于根系，$ a_ {n-1} $和$ b_n $这些贝塞尔过程与$β$ -HERMITE和$β$ -Laguerre集合相关。此外，对于冷冻情况$ k = \ infty $，这些过程将其退化为确定性或纯跳跃过程。我们将发电机用于A和B类型的Bessel和Dunkl过程，并得出Wigner的半圆的类似物和Marchenko-Pastur限制限制定律，用于$ n \ to \ Infty $，用于使用宽恕，该粒子的经验分布，通过使用释放进行置换。特别是，对于B型的dunkl过程，出现了$ \ Mathbb r $上的新的非对称半圆形限制分布。我们的结果表明，限制度量的形式已经完全取决于冷冻过程。此外，在冷冻情况下，我们的方法分别为Hermite和Laguerre多项式的零的经验度量提供了半圆和Marchenko-Pastur限制定律的新简单证明。
We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_N$ these Bessel processes are related with $β$-Hermite and $β$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type B new non-symmetric semicircle-type limit distributions on $\mathbb R$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively.
