Speaker: Yimin Xiao

Title: Collision of Eigenvalues of Random Matrices with Gaussian Random Field Entries

Abstract: Let $\xi = \{\xi(t): t \in \R_+^N \}$ be a centered Gaussian random field  and let $\{\xi_{i,j}, \eta_{i,j}:i,j \in \mathbb N\}$

be a family of independent copies of $\xi$. For $\beta \in \{1,2\}$ and a fixed integer  $d \ge 2$, consider the $d\times d$

matrix-valued process $X^{\beta} = \{X_{i,j}^{\beta}(t); t \in \R_+^N, 1 \le i,j \le d\}$ with entries given by

\begin{align} \label{def-entries}

X_{i,j}^{\beta}(t) =

\begin{cases}

\xi_{i,j}(t) + \iota \I_{[\beta = 2]} \eta_{i,j}(t), & i < j; \\

\sqrt{2} \xi_{i,i}(t), & i=j; \\

\xi_{j,i}(t) - \iota \I_{[\beta = 2]} \eta_{j,i}(t), & i > j,

\end{cases}

\end{align}

where $\iota := \sqrt{-1}$ is  the imaginary unit.  Thus, for every $t \in  \R_+^N$, $X^{\beta}(t)$ is a real symmetric matrix for $\beta=1$ and a  complex Hermitian matrix for $\beta=2$.  This setting is one of the extensions of the seminal work of Dyson (1962).

Jaramillo and Nualart (2020) provided a necessary condition and a sufficient condition for the collision of eigenvalues of $X^{\beta}$. Song et al (2021) extended their results to the case where $k$ eigenvalues collide with $2\le k\le d$ and determined the Hausdorff dimension of the set of collision times. However, in the case of ``critical dimension", the problem whether the eigenvalues of $X^{\beta}$ collide or not was left open. 

We solve this problem by extending Talagrand's covering argument. More specifically, let $X= \{X(t), t \in \R^N\}$ be a centered Gaussian random field with values in $\R^d$ satisfying certain conditions and let $F \subset \R^d$ be a Borel set. We provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P\big( X(t) \in F \hbox{ for some }  t \in \R^N \backslash\{0\}\big) = 0$.

Our new condition is related to the upper Minkowski dimension of $F$ and is applicable to a variety of examples of Gaussian random fields including random matrices with Gaussian random field entries. 

This talk is mainly based on a joint paper with  Cheuk-Yin Lee, Jian Song, and Wangjun Yuan.