The average singular value of a complex random matrix decreases with dimension

We obtain a recurrence relation in d for the average singular value α(d) of a complex valued d × d matrix 1 √ d X with random i.i.d., N (0, 1) entries, and use it to show that α(d) decreases monotonically with d to the limit given by the Marchenko-Pastur distribution. The monotonicity of α(d) has been recently conjectured by Bandeira, Kennedy and Singer in their study of the Little Grothendieck problem over the unitary group Ud [4], a combinatorial optimization problem. The result implies sharp global estimates for α(d), new bounds for the expected minimum and maximum singular values, and a lower bound for the ratio of the expected maximum and the expected minimum singular value. The proof is based on a connection with the theory of Turán determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.