Markov property of determinantal processes with extended sine, Airy, and Bessel kernels
Markov property of determinantal processes with extended sine, Airy, and Bessel kernels
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2011-06
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通讯作者：
M. Katori;H. Tanemura
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作者：
M. Katori;H. Tanemura
当粒子数量有限时，非碰撞布朗运动（戴森模型）和非碰撞平方贝塞尔过程是任何确定性初始配置$\xi=\sum_{j \in \Lambda} \delta_{x_j}$的决定性扩散过程，在某种意义上，任何多时间相关函数都是由与相关核相关的行列式给出的，该行列式由$\supp \xi$中有零的整个函数$\Phi$指定。使用这些完整的函数$\Phi$，我们定义了称为$\Phi$ -moderate拓扑的新拓扑。然后在$\Phi$ -中等拓扑结构中构造了三个无限维的行列式扩散过程，作为有限维分布意义上有限粒子数的行列式扩散过程序列的极限，使得概率分布相对于初始构型$\xi$和$\xi(\R)=\infty$是连续的。我们证明了我们的三个无限粒子系统分别是具有扩展正弦核、贝塞尔核和Airy核的行列式过程的版本，它们相对于随机矩阵理论中研究的高斯酉系综特征值分布的体积尺度极限和软边缘尺度极限所得到的行列式点过程，以及手性高斯酉系综特征值分布的硬边缘尺度极限是可逆的。然后证明了三个无限维行列式过程的马尔可夫性。
When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $\xi=\sum_{j \in \Lambda} \delta_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $\Phi$ having zeros in $\supp \xi$. Using such entire functions $\Phi$, we define new topologies called the $\Phi$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, as the limits of sequences of determinantal diffusion processes with finite numbers of particles in the sense of finite dimensional distributions in the $\Phi$-moderate topologies, so that the probability distributions are continuous with respect to initial configurations $\xi$ with $\xi(\R)=\infty$. We show that our three infinite particle systems are versions of the determinantal processes with the extended sine, Bessel, and Airy kernels, respectively, which are reversible with respect to the determinantal point processes obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied in the random matrix theory. Then Markovianity is proved for the three infinite-dimensional determinantal processes.