  Studies of fixation dynamics in Markov processes predominantly focus on the
mean time to absorption. This may be inadequate if the distribution is broad
and skewed. We compute the distribution of fixation times in one-step
birth-death processes with two absorbing states. These are expressed in terms
of the spectrum of the process, and we provide different representations as
forward-only processes in eigenspace. These allow efficient sampling of
fixation time distributions. As an application we study evolutionary game
dynamics, where invading mutants can reach fixation or go extinct. We also
highlight the median fixation time as a possible analog of mixing times in
systems with small mutation rates and no absorbing states, whereas the mean
fixation time has no such interpretation.
