各向异性Young图和具有Jack参数的infinite-dimensional扩散过程
Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter
摘要：
我们构造了一个家庭的马尔可夫过程的连续样本轨迹上的无限维空间，托马斯单纯形。该族依赖于三个连续参数，其中之一Jack参数类似于随机矩阵理论中的beta参数。该过程产生于某些有限马尔可夫链的标度极限转移，所谓的上-下链与杰克边重数的杨图。每一个极限马氏过程都是遍历的，其平稳分布是一个对称化测度。过程的无穷小生成元被显式地计算出来;作为对称化测度上的L^2空间中的自伴算子，生成元具有被显式描述的纯离散谱。对于Jack参数的特殊值1，极限马尔可夫过程与Borodin和作者（Prob. Theory Rel. Fields 144（2009），281- 318; arXiv：0706.1034）。在杰克参数为0的极限中，我们的过程族退化为金曼单形上的单参数扩散族，这是很久以前由布泽尔和库尔茨在与某些群体遗传学模型有关的研究中发现的。该文件的技术基本上是代数。主要的计算是在与杰克参数的移位对称函数的代数，并依赖于各向异性的杨图由于Kerov的概念。
英文摘要：
We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Young graph with the Jack edge multiplicities. Each of the limit Markov processes is ergodic and its stationary distribution is a symmetrizing measure. The infinitesimal generators of the processes are explicitly computed; viewed as selfadjoint operators in the L^2 spaces over the symmetrizing measures, the generators have purely discrete spectrum which is explicitly described. For the special value 1 of the Jack parameter, the limit Markov processes coincide with those of the recent work by Borodin and the author (Prob. Theory Rel. Fields 144 (2009), 281--318; arXiv:0706.1034). In the limit as the Jack parameter goes to 0, our family of processes degenerates to the one-parameter family of diffusions on the Kingman simplex studied long ago by Ethier and Kurtz in connection with some models of population genetics. The techniques of the paper are essentially algebraic. The main computations are performed in the algebra of shifted symmetric functions with the Jack parameter and rely on the concept of anisotropic Young diagrams due to Kerov.
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