Suppose you have a one dimensional random walk, with a wall at $S=0$ and a sink at $S=n$. The walk is biased so the odds of moving down vs moving up are $b:1$. More concretely, the transition graph is:
$\substack{⟳ \\ q} S_0 \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} S_1 \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} S_2 \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} S_3 \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} S_4 \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} ... \substack{p \\ \longrightarrow \\ \longleftarrow \\ q} S_{n-1} \substack{p \\ \longrightarrow} S_n \substack{⟲ \\ 1}$
where $p=\frac{1}{b+1}$ and $q = \frac{b}{b+1}$.
The most significant eigenvector for the walk, the steady-state vector with eigenvalue 1, is the vector $\langle 0,0,0,0,..., 0, 0, 1 \rangle$ with all probability sunk into $S_n$.
I want to compute the eigenvector with the second-largest eigenvalue.
Based on the steady state when there's no wall and no sink, I think that the second eigenvector will be approximately proportional to $\langle b^n,b^{n-1}, b^{n-2}, b^{n-3}, ..., b^2, b^1, ? \rangle$ when $n$ and $b$ are large and that the eigenvalue will be quite close to 1. I also think that, when $b=2$, the vector $\langle 2^n-1, 2^{n-1}-1, 2^{n-2}-1, ..., 2^2-1,2^1-1, ? \rangle$ is a better proportional approximation for large $n$. At least, that's what simulation seems to show.
Because the characteristic polynomial of the walk's transition matrix is quite large, I'm not sure how to get a simple answer (if one even exists) or how to validate or reject the above approximations that honestly I just guessed at.
Any useful tips for solving this?