Conisder the following dimension stochastic matrix,  \begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ \end{bmatrix} with $p,q>0$ and $p+q=1$.

To control mixing time, I am interested in its spectral gap. Let $n$ be the number of rows (e.g., $n=5$ in the example above). The characteristic polynomial is  $$x^n - px^{n-1} - q.$$ How to argue that the spectral gap decay polynomially in terms of $n$ (rather than of exponentially)?

Consider the following dimension stochastic matrix, 

\begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ \end{bmatrix}

with $p,q>0$ and $p+q=1$.

To control mixing time, I am interested in its spectral gap. Let $n$ be the number of rows (e.g., $n=5$ in the example above). The characteristic polynomial is 

$$x^n - px^{n-1} - q.$$

How to argue that the spectral gap decays polynomially in terms of $n$ (rather than exponentially)?