Mixing times of Markov chains (M2 - 2023)
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 Université Paris-Dauphine on Mondays, 13:45-17:00, starting from February 6.
- In order to take the exam, it is mandatory to register for the course by sending an e-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 10, 14:00-17:00.
- 06.02 - General framework,
- 13.02 - Probabilistic aspects,
- 20.02 - Spectral aspects,
- 06.03 - Geometric aspects 1,
- 13.03 - Geometric aspects 2,
- 20.03 - Variational aspects 1,
- 27.03 - Variational aspects 2,
- 03.04 - Advanced examples,
- 10.04 - Final exam.