Let $A_n$ be the matrix product of $n$ i.i.d. N-by-N random complex matrices. The matrix distribution is not fixed and can be tuned to suit specific solution if needed, as long as it's not too "special". One possible choice is with i.i.d. uniform or Gaussian entries with 0 mean. What I want to know is the asymptotic bahavior of the ratio $r_n$ between the second greatest singular value and the greatest of $A_n$ with respect to $n$. (Dimension of the matrices $N$ is fixed and may not be large)

This question comes from a physics problem and we actually want to prove that this ratio will decay exponentially with respect to $n$: $r_{n\rightarrow\infty}\propto exp({-\alpha n}) $, with probability 1, which is supported by some numerical simulation.