轨道测度与交错行列式点过程

轨道测度、随机矩阵理论与交错行列式过程

Orbit measures, random matrix theory and interlaced determinantal processes

描述了紧群的表示与埃尔米特矩阵的某些不变系综之间的联系。我们专注于两种类型的不变系综，扩展高斯和拉盖尔酉系综。我们研究他们使用的投影和卷积不变的概率测度的伴随轨道的紧凑李群。这些措施所描述的半经典近似涉及张量和限制mulltiplicities。我们发现，他们中的一大类是行列式。

循环上的交错过程

Interlaced processes on the circle

当两个马尔可夫算子交换时，它表明我们可以耦合相应过程之一的两个副本。我们明确地构建了一些这种类型的耦合交换家庭的马尔可夫过程的共轭类的酉群，使用的动力学规则的启发RSK算法。我们这样做的动机是开发一个平行的程序，在循环，一些最近发现的随机矩阵理论之间的反射和条件系统的粒子线。我们考虑的一个马尔可夫链在无限圆柱上的“珠状构型”上产生了一族吉布斯测度。我们表明，这些措施具有行列式结构，并计算相应的时空相关核。

Universal edge fluctuations of discrete interlaced particle systems

We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical d…

Asymptotic geometry of discrete interlaced patterns: Part I

A discrete Gelfand-Tsetlin pattern is a configuration of particles in Z^2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc, and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform p…