The Anti-Symmetric GUE Minor Process

来自变形 GUE 子式过程的蜂巢

Hives from deformed GUE minor processes

从两个独立对角变形的 GUE 矩阵各自的变量经子式过程形成双蜂巢并应用八面体递推，在匹配条件下，证明所得蜂巢律在相对熵上接近 GUE 蜂巢律。

完美匹配在铁路场图上的独立GUE次过程

Independent GUE minor processes of perfect matchings on rail-yard graphs

研究铁路场图上完美匹配的分布收敛性，通过Schur函数公式分析得出独立GUE次过程的谱特性。

Perturbed GUE Minor Process and Warren's Process with Drifts

We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand-Tsetlin pattern that also appears in a generalization of Warren's process, in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions. Our results generalize the known results for the zero drift situation.

Tacnode GUE-minor Processes and Double Aztec Diamonds

We study random domino tilings of a Double Aztec diamond, a region consisting of two overlapping Aztec diamonds. The random tilings give rise to two discrete determinantal point processes called the K-and L-particle processes. The correlation kernel of the K-particles was derived in Adler, Johansson and van Moerbeke (2011), who used it to study the limit process of the K-particles with different weights for horizontal and vertical dominos. Let the size of both, the Double Aztec diamond and the overlap, tend to infinity such that the two arctic ellipses just touch; then they show that the fluct…

Plane partitions with 2-periodic weights

We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The turning points that appear in the homogeneous case split in our model into pairs of turning points macroscopically separated by a "semi-frozen" region. As a result the point process at a turning point is not the GUE minor process, but rather a pair of GUE minor processes, non-trivially correlated. We also study an intermediate regime when the weights are periodic …

双周期Aztec钻石中的GUE-角过程

GUE-corners process in two-periodic Aztec diamonds

该研究探讨修改Aztec钻石构型的概率测度后，其多米诺骨牌的概率密度函数在大尺寸极限下是否仍收敛到GUE-角过程，在两种权重情形下证明了收敛性并给出精确分布表达式。

The Dyson Brownian minor process

Consider an $n\times n$ Hermitean matrix valued stochastic process $\{H_t\}_{t\geq 0}$ where the matrix elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect. In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the $k\times k$ in the upper left corner of $H_t$. If you project this process to a space-like path it is a determinantal …

Wigner子式过程中的去相关转变

Decorrelation transition in the Wigner minor process

本文研究了Wigner子式过程的去相关转变，证明了Forrester和Nagao针对GUE子式过程在临界区域$k\sim N^{2/3}$给出的非平凡联合相关函数公式具有普适性，并对亚临界和超临界区域的特征值及特征向量重叠进行了完整分析。

Eigenvector distribution in the critical regime of BBP transition

In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik-Ben Arous-Péché (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition (arXiv:math/0403022). The derivation of the distribution makes use of the recently re-discovered eig…

Limits of determinantal processes near a tacnode

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter $\eps> 0$. The domain has two cusps, one pointing up and one pointing down. In the limit $\varepsilon\downarrow 0$ the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime $\varepsilon \downarrow 0$. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

随机矩阵的分数对数定律

Law of fractional logarithm for random matrices

该研究完全一般化地证明了随机矩阵极端特征值的Paquette-Zeitouni分数对数定律，验证了相关猜想，结果适用于任意Wigner子式过程及两种对称类，通过鞅技术和动力学方法分别刻画了不同k尺度下矩阵最大特征值与子式的相关性强弱。

八面体递推的极限（带GUE边界数据）

Limit of the octahedron recurrence with GUE boundary data

研究带GUE边界数据的八面体递推，证明其缩放后随机场收敛于确定性Lipschitz函数，并通过完美匹配公式给出表面张力变分刻画。