arXiv subjects

Alexander Tikhomirov

Publications and source records attributed to Alexander Tikhomirov.

22 records · Page 2Linked to original sources

arXiv2010

On the asymptotic distribution of the singular values of powers of random matrices

We consider powers of random matrices with independent entries. Let $X_{ij}, i,j\ge 1$, be independent complex random variables with $\E X_{ij}=0$ and $\E |X_{ij}|^2=1$ and let $\mathbf X$ denote an $n\times n$ matrix with $[\mathbf X]_{ij}=X_{ij}$, for $1\le i, j\le n$. Denote by $s_1^{(m)}\ge...\ge s_n^{(m)}$ the singular values of the random matrix $\mathbf W:={n^{-\frac m2}} \mathbf X^m$ and define the empirical distribution of the squared singular values by $$ \mathcal F_n^{(m)}(x)=\frac1n\sum_{k=1}^nI_{\{s_k^{(m)}^2\le x\}}, $$ where $I_{\{B\}}$ denotes the indicator of an event $B$. We prove that under a Lindeberg condition for the fourth moment that the expected spectral distribution $F_n^{(m)}(x)=\E \mathcal F_n^{(m)}(x)$ converges to the distribution function $G^{(m)}(x)$ defined by its moments $$ α_k(m):=\int_{\mathbb R}x^k\,d\,G(x)=\frac {1}{mk+1}\binom{km+k}{k}. $$

math.PR↗

arXiv2010

The circular law for random matrices

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.

math.PR↗

arXiv2010

Asymptotic distribution of singular values of powers of random matrices

Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product $\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is Hermitian positive semi-definite. Let $λ_1,λ_2,...,λ_N$ be eigenvalues of $\Xb^m{\Xb^*}^m$ (or squared singular values of the matrix $\Xb^m$). In this paper we find the asymptotic distribution function \[ G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \] of the empirical distribution function \[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{λ_k \leq x\}}}, \] where $\mathbb{I} \{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy \[ M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \] In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967.

math.PR↗

arXiv2000

Sur le morphisme de Barth

Let ${\rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_{1}=0,c_{2}=n$. The curve $β_{\rm F}$ of jumping lines of ${\rm F}$, in the dual projective plane, has degree $n$. Let ${\rm M}_{n}$ be the moduli space of equivalence classes of semi-stables sheaves of rank 2 and Chern classes $(0,n)$ on the projective plane and ${\cal C}_{n}$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $$β: {\rm M}_{n}\longrightarrow{\cal C}_{n}$$ associates the point $β_{\rm F}$ to the class of the sheaf ${\rm F}$. We prove that this morphism is generically injective for $n\geq 4.$ The image of $β$ is a closed subvariety of dimension $4n-3$ of ${\cal C}_{n}$; as a consequence of our result, the degree of this image is given by the Donaldson number of index $4n-3$ of the projective plane.

math.AG↗