Feel free to choose a paper related to MCMC and your interests. Below
are some suggestions but you are welcome to ignore and choose your own topic.
You might check this paper by Diaconis for some additional ideas.
Random walks on small world networks.
Project idea: Run simulations for the directed version to
analyze its stationary distribution.
Phase transitions: For models such as the Ising model and hard-core model (on weighted independent sets) run simulations of the Gibbs sampler to
test the experimental mixing time, and how it compares to phase transitions
in the model. For example, see
and this Propp-Wilson paper.