Università Roma Tre

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

Main reference:
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Additional reference (in Italian)
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi