We state and prove a version of Szeg\H{o}'s first limit theorem for multi-Toeplitz operators acting in the full Fock space in $d\geq 2$ letters. In particular we show that the eigenvalue distributions of truncated multi-Toeplitz operators converge to a limiting distribution. We compute this limiting distribution and its associated entropy in some special cases. Even in the simplest examples the limiting distribution is quite different from the classical case (corresponding to $d=1$); in these examples we obtain a purely atomic measure, which admits a probabilistic interpretation in connection with a percolation process associated to a simple tridiagonal random matrix model.
We study the eigenvalues of the random Laplacian matrix $D - A$, where $A$ is a Wigner matrix with sub-Gaussian entries and the diagonal matrix $D$ contains the row sums of $A$. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering an...
We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenva...
We study the empirical eigenvalue distribution of a standardized up Laplacian of the Linial-Meshulam model $Y_{q+1}(n, p)$. We first give a definition of a Gaussian--semicircle law: $\mathcal{N}(0, \sigma^2)\boxplus \operatorname{SC}(s\sigma^2)$, and give a combinatorial formula of its moments in terms of pairing parti...
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}...
Little is known about the dynamics of Toeplitz operators on the Bergman space, and even in the case of harmonic symbols, few results are available. In this paper, we study the dynamical properties of Toeplitz operators with symbols of the form $a \bar{z} + p (z)$ on the Bergman space, where $a\neq 0$ and $p$ is analyti...
Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $\Gamma$-independent semicircular elements. This characterizes their limiting s...
Afonso S. Bandeira, Dmitriy Kunisky, Petar Nizic-Nikolac et al.· 0 citations
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