  We investigate the application of Weak Poincar\'e Inequalities (WPI) to
Markov chains to study their rates of convergence and to derive complexity
bounds. At a theoretical level we investigate the necessity of the existence of
WPIs to ensure \mathrm{L}^{2}-convergence, in particular by establishing
equivalence with the Resolvent Uniform Positivity-Improving (RUPI) condition
and providing a counterexample. From a more practical perspective, we extend
the celebrated Cheeger's inequalities to the subgeometric setting, and further
apply these techniques to study random-walk Metropolis algorithms for
heavy-tailed target distributions and to obtain lower bounds on pseudo-marginal
algorithms.
