Consider a discrete state space $\mathcal{X}$. The expander Chernoff inequality gives subgaussian concentration for the sample mean $\frac1n \sum_{t=1}^n f(X_t)$ for some function $f : \mathcal{X} \to [0,1]$, where $X_t$ follows a discrete time stationary Markov chain. The variance parameter of the subgaussian concentration itself is proportional to the spectral gap of the chain. A small spectral gap would be implied by bottlenecks in the Markov chain (Cheeger's inequality) which would in turn make convergence slow.
Would I be right to think that the existence of bottlenecks would only be problematic if the chain is stationary? By this I mean: assume that the Markov chain is always started from a fixed state $X$, and compare $\frac1n \sum_{t=1}^n f(X_t)$ with its expectation, $\frac1n \mathbb{E}_{\substack{X_1 = X,\\ X_{i+1} \sim P(\cdot | X_i)}} \left[\sum_{t=1}^n f(X_i) \right]$. Can one expect subgaussian concentration with a variance parameter that does not depend on the spectral gap of the chain?
In the extreme case, where the Markov chain has two disjoint components, the stationary chain will never converge, since it may never reach the vertices in the other component. However, when started in a fixed state, the 2nd disjoint component can be thrown away since it appears neither in the observed sample paths, nor in the expectations.
