Suppose we have an aperiodic matrix $A_t$ that has entries that are either $0$ or are positive integer powers of $t$, i.e. we could have $$A_t = \begin{pmatrix} 0 & t & t^2\\ t & t^2 & 0\\ t & 0 & t \end{pmatrix}$$ for example.

Suppose $t>0$ and let $\Lambda(t)$ denote the unique, real, simple maximal eigenvalue of $A_t$ guaranteed by the Perron-Frobenius Theorem. If we consider the function $$f(t) = \log\Lambda(e^t)$$ then it is possible to show using a variational principle and perturbation theory that $f(t)$ is increasing, convex and analytic (this is non-trivial!) with uniformly bounded (for $t\in\mathbb{R}$) first derivative. In particular the limits $$\lim_{t\to\infty} \frac{f(t)}{t} = \alpha_1 \ \ \text{and} \ \ \lim_{t\to - \infty} \frac{f(t)}{t} = \alpha_2$$ both exist and are finite. My question is the following: can we calculate the error term associated to these limits? That is, can we find $g(t)$ such that $$f(t) = \alpha_1 t + O(g(t))$$ as $t\to\infty$ for example?

Edit: I would ideally like to show that the error is $O(1)$. This is equivalent to the fact that $\Lambda(e^t)$ grows purely exponentially, i.e. there exists $C, \lambda \ge 1$ such that $$\frac{1}{C} \lambda^t \le \Lambda(e^t) \le C \lambda^t$$ for all $t$ sufficiently large. An equivalent inequality should also hold for $-t$ sufficiently large.

Any thoughts/insights would be greatly appreciated - thanks!