Events (We may skip the seminar in case it overlaps with some other academic events.)
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Abstract: In this talk we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of dimension $n$ and order-$r$, drawn from the density \[ f(\mathcal{G}) = \frac{1}{Z_r(n)} \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg).\]
We consider contractions of the form $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain a Baik-Ben Arous-P\'{e}ch\'{e} phase transition for the largest and the smallest eigenvalues of such contractions at $r = 3$. We also show that the extreme eigenvectors contain non-trivial information about $\mathbf w$. In fact, in the regime $1 \ll r \ll n$, there are two vectors, one of which is perfectly aligned with $w$. We also obtain some results on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ in the case $r = 4$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Further, it turns out that there are no outlier eigenvalues when $\langle \mathbf{u}, \mathbf{v}\rangle = o(1)$. This talk is based on a joint work with Soumendu Sundar Mukherjee.
Abstract: For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz–Hausdorff theorem. The largest value in this convex set is known as the numerical radius, which is often used to study the convergence rate of iterative methods for solving linear systems. In this talk, we will introduce a recent result on the asymptotic behavior of the numerical radius of a large-dimensional, complex, non-Hermitian random matrix and its elliptic variants. For the former, remarkably, the radius can be represented as the extremum of a stationary Airy-like process, which undergoes a correlation-decorrelation transition from a small to a large time scale. Based on this transition, we obtain the precise first and second order terms of the numerical radius. In the elliptic case, we show that the fluctuation of the numerical radius reduces to the maximum or minimum of two independent Tracy-Widom variables. Based on joint work with Giorgio Cipolloni.