Central Limit Theorem for Linear Statistics of Eigenvalues of Band Random Matrices

The goal of this paper is to prove the Central Limit Theorem for linear statistics of the eigenvalues of real symmetric band random matrices with independent entries. First, we define a real symmetric band random matrix. Let {bn} be a sequence of integers satisfying 0 ≤ bn ≤ n/2 such that bn → ∞ as n → ∞. Define dn(j, k) := min{|k − j|, n− |k − j|}, (1.1) In := {(j, k) : dn(j, k) ≤ bn, j, k = 1, . . . , n} and (1.2) I n := {(j, k) : (j, k) ∈ In, j ≤ k}. In particular, dn has the following natural interpretation: if the first n positive integers are evenly spread out on a circle of radius n 2π , then dn(j, k) is the distance between the integers j and k. The quantity bn will be the radius of a band of our random matrix. In other words, all entries of the matrix with j, k / ∈ In are going to be zero. We define a real symmetric band random matrix