LAUSR

Abstract We investigate distribution of eigenvalues of growing size Toeplitz matrices [ a n + k − j ] 1 ≤ j , k ≤ n as n → ∞ , when the entries { a j } are “smooth” in the sense, for example, that for some α > 0 , a j − 1 a j + 1 a j 2 = 1 − 1 α j ( 1 + o ( 1 ) ) , j → ∞ . Typically they are Maclaurin series coefficients of an entire function. We establish that when suitably scaled, the eigenvalue counting measures have limiting support on [ 0 , 1 ] , and under mild additional smoothness conditions, the universal scaled and weighted limit distribution is | π log ⁡ t | − 1 / 2 d t on [ 0 , 1 ] .