Wigner matrix ($X_{ij}=N^{-1/2}{Y}_{ij}$)

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.