How many times must one shuffle a deck of 52 cards? This course is a self-contained introduction to the modern theory of mixing times of Markov chains. It consists of a guided tour through the various methods for estimating mixing times, including couplings, spectral analysis, discrete geometry, and functional inequalities. Each of those tools is illustrated on a variety of examples from different contexts: interacting particle systems, card shufflings, random walks on groups, graphs and networks, etc. Finally, a particular attention is devoted to the celebrated cutoff phenomenon, a remarkable but still mysterious phase transition in the convergence to equilibrium of certain Markov chains.

- All lectures take place at UniversitĂ© Paris-Dauphine on Mondays, 13h45-17h00, starting from February 7.
- In order to take the exam, it is mandatory to register for the course, by sending me ane-mail before March 1 with your name and the name of the university in which you are registered for your Master.
- The exam will take place on April 11, 14h-17h.

- Lecture notes (in construction, comments welcome!)
- Exam 2021 and solution
- Exam 2020 and solution
- Markov Chains and Mixing Times (D. Levin, Y. Peres & E. Wilmer)
- Mathematical Aspects of Mixing Times in Markov Chains (R. Montenegro & P. Tetali)
- Mixing Times of Markov Chains: Techniques and Examples (N. Berestycki)
- Reversible Markov Chains and Random Walks on Graphs (D. Aldous & J. Fill)

- Universality of cutoff for exclusion with reservoirs (Interacting particle systems)
- Cutoff for non-negatively curved Markov chains (Curvature, cutoff).
- Cutoff for the mean-field zero-range process (Interacting particle systems).
- Mixing time of the adjacent walk on the simplex (Continuous state space).
- Cutoff on all Ramanujan graphs (Random walks on graphs, spectral aspects).
- Cutoff phenomena for random walks on random regular graphs (Random walks on random graphs).
- Universality of cutoff for graphs with an added random matching (Random walks on random graphs).
- Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability (Coupling, Ising model).
- An exposition to information percolation for the Ising model (Coupling, Ising model).
- Trailing the Dovetail Shuffle to its Lair (Card shuffling).
- Path coupling without contraction (Path coupling, curvature).
- The Proppâ€“Wilson algorithm (Perfect sampling, algorithms).
- Mixing times of lozenge tilings and card shuffling Markov chains (Coupling, spectral aspects).