Dyson's Brownian motions, intertwining and interlacing

Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains -- an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding constru…

Intertwinings of beta-Dyson Brownian motions of different dimensions

We show that for all positive beta the semigroups of beta-Dyson Brownian motions of different dimensions are intertwined. The proof relates beta-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption beta>1 in [GS]. The corresponding results for beta-Dyson Ornstein-Uhlenbeck processes are also presented.

Consecutive Minors for Dyson's Brownian Motions

In 1962, Dyson introduced dynamics in random matrix models, in particular into GUE (also for beta=1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the ge…

大型高斯矩阵的交错特征向量

Interlacing Eigenvectors of Large Gaussian Matrices

本文研究大型高斯矩阵主子式特征向量与全矩阵特征向量的渐近重叠，通过Burgers型演化方程计算极限重标度均方重叠，在GOE情况下简化公式并发现特征向量交错现象，方法基于分析Dyson布朗运动下的特征向量流。

Interacting particle systems at the edge of multilevel Jack processes

We consider a multilevel continuous time Markov chain $X(s;N) = (X_i^j(s;N): 1 \leq i \leq j \leq N)$, which is defined by means of Jack symmetric functions and forms a certain discretization of the multilevel Dyson Brownian motion. The process $X(s;N)$ describes the evolution of a discrete interlacing particle system with push-block interactions between the particles, which preserve the interlacing property. We study the joint asymptotic separation of the particles at the right edge of the ensemble as the number of levels and time tend to infinity and show that the limit is described by a cer…

Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime

We consider a one-dimensional system of Brownian particles that repel each other through a logarithmic potential. We study two formulations for the system and the relation between them. The first, Dyson's Brownian motion model, has an interaction coupling constant determined by the parameter beta > 0. When beta = 1,2 and 4, this model can be regarded as a stochastic realization of the eigenvalue statistics of Gaussian random matrices. The second system comes from Dunkl processes, which are defined using differential-difference operators (Dunkl operators) associated with finite abstract vector …