Jaewhi Park | Searcharxiv

arXiv2023

Tracy-Widom limit for free sum of random matrices

We consider fluctuations of the largest eigenvalues of the random matrix model $A+UBU^{*}$ where $A$ and $B$ are $N \times N$ deterministic Hermitian (or symmetric) matrices and $U$ is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the Tracy-Widom distribution, under mild assumptions on $A$ and $B$ to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of the Green function along the Dyson Brownian motion starting from the matrix $A + UBU^{*}$ and ending at time $N^{-1/3+χ}$. As a byproduct of our proof, we also prove an optimal local law for the Dyson Brownian motion up to the constant time scale.

math.PR↗

arXiv2020

Extremal eigenvalues of sample covariance matrices with general population

We consider the eigenvalues of sample covariance matrices of the form $\mathcal{Q}=(Σ^{1/2}X)(Σ^{1/2}X)^*$. The sample $X$ is an $M\times N$ rectangular random matrix with real independent entries and the population covariance matrix $Σ$ is a positive definite diagonal matrix independent of $X$. Assuming that the limiting spectral density of $Σ$ exhibits convex decay at the right edge of the spectrum, in the limit $M, N \to \infty$ with $N/M \to d\in(0,\infty)$, we find a certain threshold $d_+$ such that for $d>d_+$ the limiting spectral distribution of $\mathcal{Q}$ also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of $\mathcal{Q}$ are determined by the order statistics of the eigenvalues of $Σ$, and in particular, the limiting distribution of the largest eigenvalue of $\mathcal{Q}$ is given by a Weibull distribution. In case $d<d_+$, we also prove that the limiting distribution of the largest eigenvalue of $\caQ$ is Gaussian if the entries of $Σ$ are i.i.d. random variables. While $Σ$ is considered to be random mostly, the results also hold for deterministic $Σ$ with some additional assumptions.

math.PR↗