Random matrix

A random matrix is a matrix whose entries are random variables. Its eigenvalues form a random configuration, and its empirical spectral measure records their distribution with equal mass on each eigenvalue. Entry assumptions and normalization determine the spectral regime.

For an $N$-by-$N$ Hermitian matrix, the empirical spectral measure puts mass $1/N$ at each eigenvalue, counted with multiplicity. It is a probability measure. Integrating a test function against it gives the average of that function evaluated at the eigenvalues.

A real Wigner matrix is a real symmetric matrix whose upper off-diagonal entries are independent centered copies of a fixed random variable, scaled to have variance of order $1/N$. Diagonal conventions may vary and should be specified separately. A common version has variance-one unscaled off-diagonal entries and either zero diagonal entries or independent diagonal entries with a bounded second moment.

Truncate the unscaled entries of a Wigner matrix at a fixed threshold and subtract their expected values. The difference between original and truncated entries is $N^{-1/2}(Z_C-\mathbb{E}{Z}_C)$ with $Z_C=Y{1}_{|Y|\ge C}$, so its variance is bounded by $N^{-1}\mathbb{E}[Y^2{1}_{|Y|\ge C}]$. The centering term cannot be discarded for an asymmetric entry law.

For a fixed mean-zero variance-one real entry law, centered truncation of a Wigner matrix satisfies $\mathbb{E}[N^{-1}\mathrm{Tr}(X-\hat{X}{)}^2]\le t(C)$, where $t(C)=\mathbb{E}[Y^2{1}_{|Y|\ge C}]\to 0$. The spectral Lipschitz bound from Frobenius distance and Markov inequality give probability at most $t(C)/{\epsilon }^2$ for a spectral test-function discrepancy exceeding $\epsilon$. Thus $t(C)<{\epsilon }^3$ works uniformly in matrix size. A varying entry-law family requires uniform second-moment tail decay instead.

A complex Wishart matrix is the Gram matrix of an $n$-by-$m$ matrix of independent centered complex Gaussian random variables with an indicated variance convention. It is a positive semidefinite matrix. In the standard convention with joint entry density proportional to $e^{-\mathrm{Tr}{X}^{\ast }X}$ and $n\ge m$, its eigenvalue density contains the weight $x^{n-m}{e}^{-x}$ and a squared Vandermonde determinant.

The Laguerre unitary ensemble is the positive-eigenvalue ensemble with joint density proportional to ${\prod }_{i<j}(x_j-x_i{)}^2{\prod }_i{x}_i^a{e}^{-x_i}$, for $a>-1$. Integer $a=n-m\ge 0$ is realized by a complex Wishart matrix. The Laguerre projection kernel expresses its correlation functions of a point process as determinants.