Abstract
We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.
Highlights
Random matrices were introduced in pioneering works by Wishart [43] and Wigner [42] for applications in mathematical statistics and nuclear physics, respectively
Wigner argued that the energy level statistics of large atomic nuclei could be described by the eigenvalues of a large Wigner matrix, i.e., a hermitian matrix H =Ni,j=1 with centered, identically distributed and independent entries
He proved that the empirical spectral measure converges to the semicircle law as the dimension of the matrix N goes to infinity
Summary
Random matrices were introduced in pioneering works by Wishart [43] and Wigner [42] for applications in mathematical statistics and nuclear physics, respectively. If T is diagonal, they are generalizations of sample covariance matrices as T ZZ∗T ∗ = XX∗ and the elements of X = T Z are independent With this definition, all entries within one row of X have the same variance since sij = E|xij |2 = (T T ∗)ii, i.e., it is a special case of our random Gram matrix. In all previous works concerning general Gram matrices, the spectral parameter z was fixed, in particular Im z had a positive lower bound independent of the dimension of the matrix This positive imaginary part provided the necessary contraction factor in the fixed point argument that led to the existence, uniqueness and stability of the solution to the Dyson equation, (1.3).
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