@WillieWong not sure I understand your point. To clarify, I would like to find an asymptotic estimate of the form $I(n)\sim f(n)$, where $f(n)$ decays exponentially. Numerics suggests that $f(n)=c^{2n}/\sqrt{n}$, where $c=k-\sqrt{k^2-1}$ but I'm looking for a formal proof.

@WillieWong Yes, correct. However, I would like to explicitly characterize the decay rate of $I(n)$ as a function of parameter $n$.

@ChristianRemling yes it is not symmetric (there was a typo). However $g(A,t)$ is similar to a symmetric matrix.

@IlyaBogdanov: yes thanks! I fixed the typo. I enumerate the eigenvalues in an increasing order: $\lambda_1$ is the smallest eigenvalue of $g(A,t)$. Basically, I wonder whether there are no crossings between the eigenvalues.