For a reversible finite-state Markov chain, the second largest eigenvalue determines how fast the Markov chain converges to its stationary distribution.
A few questions (e.g, 1, 2) refer to Chapter 12 (examples starting on page 165) of the book Markov Chains and Mixing Times by Levin, Peres, and Wilmer, where the eigenvalues of some symmetric random walks are found.
Are there similar results available for asymmetric random walks? I am struggling to derive what the eigenvalues are in that case.