Asymptotic distribution of singular values of powers of random matrices

By N. Alexeev, Friedrich Götze, A. Tikhomirov. 2010

66 citations recorded by the source corpus.

Let x be a complex random variable such that $$ {\mathbf{E}}x = 0,\,{\mathbf{E}}{\left| x \right|^2} = 1 $$, and $$ {\mathbf{E}}{\left| x \right|^4} < \infty $$. Let $$ {x_{ij}},i,j \in \left\{ {1,2, \ldots } \right\} $$, be independent copies of x. Let $$ {\mathbf{X}} = \left( {{N^{ - 1/2}}{x_{ij}}} \right) $$, 1≤i,j≤N, be a random matrix. Writing X∗ for the adjoint matrix of X, consider the product XmX∗m with some m ∈{1,2,...}. The matrix XmX∗m is Hermitian positive semidefinite. Let λ1,λ2,...,λN be eigenvalues of XmX∗m (or squared singular values of the matrix Xm). In this paper, we find the asymptotic distribution function $$ {G^{(m)}}(x) = {\lim_{N \to \infty }}{\mathbf{E}}F_N^{(m)}(x) $$ of the empirical distribution function $$ F_N^{(m)}(x) = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\mathbb{I}\left\{ {{\lambda_k} \leqslant x} \right\}} $$, where $$ \mathbb{I}\left\{ A \right\} $$ stands for the indicator function of an event A. With m=1, our result turns to a well-known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457–483, 1967].

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/70f5fd3f8f02302ff2982e8fea1250ffd0e8d8de

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Generating random density matrices (2010, background): “The same Fuss–Catalan distribution describes asymptotically the statistics of singular values of s-th power of a single random Ginibre matrix [36].”