A recurrence relation for the average singular value of a complex Gaussian random matrix

In this brief note, we obtain a recurrence relation for the average singular value α (d) α ( d ) of a complex valued d× d d × d matrix 1√d X 1 d X with Gaussian entries, of the form: aligned α C( d+1) =α C( d) +R(d), aligned α C d + 1 = α C d + R ( d ) , where R ( d ) can be written as the linear combination of two integrals involving Laguerre polynomials. This reduces to showing that $$R(d)<0$$ R ( d ) < 0 (a non-trivial analysis problem) a 2016 conjecture of Bandeira, Kennedy and Singer, where it is predicted that α (d) α ( d ) decreases monotonically with d , from α C( 1) =√π /4 α C 1 = π 4 , to the d→ ∞ d → ∞ limit 8/3π 8 3 π provided by the Marchenko-Pastur distribution. While this may point to a possible path to solving the conjecture, it is still far from a solution, since the two integrals R ( d ) are non-summable hypergeometric functions, and showing that $$R(d)<0$$ R ( d ) < 0 is a non-trivial analysis problem.