The subdominant eigenvalue of a large stochastic matrix

Using intuition and computer experimentation, Bródy conjectured that the ratio of the subdominant eigenvalue to the dominant eigenvalue of a positive random matrix (with identically and independently distributed entries) converges to zero when the number of sectors tends to infinity. In this paper, we discuss the deterministic case and, among other things, prove the following version of this conjecture: if each entry of the matrix deviates from l/n by at most τ/n^1+ε, then the modulus of the subdominant root is at most τ/n^ε, where τ and ε are arbitrary positive real parameters.