Let $ (a_n) $ be a sequence of non-negative real numbers such that $\sum_{i = 0 }^\infty a_i = 1$. We assume that only a finite numbers of the $a_i$s are non zero. Let $M_d$ be the $d \times d$ symmetric (banded) Toeplitz matrix defined by:
$$ \begin{pmatrix} a_0 & a_1 & a_2 & \cdots & \cdots & \cdots \newline a_1 & a_0 & a_1 & a_2 & \cdots &\cdots \newline a_2 & a_1 & a_0 & a_1& a_2 & \dots \newline \vdots & \vdots & \vdots &\vdots & \vdots & \vdots \newline \end{pmatrix} $$
We further assume that $M_d$ is irreducible, at least for $d$ large enough.
By the Perron-Frobenius theorem $M_d$ has a (unique) eigenvector $\phi_d$ associated with the maximal eigenvalue $\lambda_d$ of $M_d$ and such that all the coefficients of $\phi_d$ are positive.
I am looking for an upper bound of the maximal ratio between two coefficients of this eigenvector, i.e.,
$$ \rho_d = \max_{0 \leq i,j \leq d-1 } \frac{\phi_d(j)}{\phi_d(i)} $$
A bound such as $\rho_d \leq C d $ where $C$ is a constant depending only on the $(a_i)$ would be ideal but any polynomial (or even larger) bound would be great. There are a few reasons that makes me think such a bound holds:
it is true for tridiagonal (i.e $a_i = 0$ if $i \geq 2 $ ) matrices as the eigenvector can be computed in this case and is of the form $\phi_d(j) = \sin(\pi \frac{j+1}{d+1}) $. This fact can be found in the book Spectral properties of banded Toeplitz matrices by Böttcher and Grudsky, chapter 12.
the matrix is "close to stochastic" (when $d$ is big) so we could expect the eigenvector to be roughly uniform.
I am not very familiar with the theory of Toeplitz matrix and I am hoping this may be elementary even though I could not find anything like it in Böttcher and Grudsky book or elsewhere.
The rest of my question provides some context and is most likely irrelevant to solve the problem. I am looking for such a bound as I am trying to find a quite sharp bound on the exit probability of a discrete random walk from an interval that is slowly growing with time . This is related to the ratio $\rho_d$ and to the asymptotic of $\lambda_d$. This asymptotic is very sharply known $ \lambda_d = 1 - \frac{\pi^2}{2 d^2} + O(\frac{1}{d^3}$) (see e.g. this paper ) by Widom.
Answering this question would also answer my previous question on Large deviation principle for small values of the maximum of the absolute value of a random walk. I can explain why in more details if needed but I doubt, it's going to be relevant.