Canonical example of Markov chain without spectral gap Is there a classical example of a Markov chain with general state space $X$ without a spectral gap? That is, a sequence of eigenvalues $\lambda_n\to1$, and continuous $f_n$ such that
$\int_X f_n(y) P(x, dy)=\lambda_n f_n(x)$ for all $x$ and $n$.
 A: Here's how I'd build one such Markov chain. I'm not sure if it would technically quality as "classical".
By definition, $X$ will be of infinite cardinality. To simplify, let's assume $X=\{x_n\}_{n\in\mathbb Z}$ to be countable (or apply the following construction to a countable subset of $X$).
Method 1: Random walks on graphs with disconnected cliques
You can build trivial examples where the multiplicity of $1$ as an eigenvalue is infinite. To do that, consider transition matrices that are block diagonal, with each block being stochastic. These model random walks on infinite graphs with infinitely many disconnected cliques (here the eigenvalue is of infinite multiplicity). The identity matrix is a degenerate transition matrix of that type.
Method 2: Triangular transition matrix
This time,  define:
$$\left\{
\begin{array}{lll}
p(x_n, x_n)&= 1-\frac 1 n\\
p(x_n, x_{n+1}) &= \frac 1 n\\
p(x_n, x_{n+k})&=0 &\text{ if }k\notin\{0, 1\}
\end{array}
\right.$$
The (infinite) transition matrix is triangular with eigenvalues that accumulate at $1$.
