A PDF copy of the course program can be found here: IMT_3800.pdf

June 14 & 17: We proved the theorem of Morris and Peres on the evolving-set method (section 17.4). The proof brought together various elements of the theory of martingales along with the material on the bottleneck ratio from chapter 7.

June 7 & 9: We proved the theorem of Jerrum and Sinclair on the existence of a (fully polynomial) randomized approximation scheme for estimating the number of proper colorings on a graph (section 14.4).

May 31: We utilized optimal transport arguments to estimate the mixing time of the Glauber dynamics for the problem of graph coloring (section 14.3).

The fourth homework assignment has been posted.

May 26: We used the theory of optimal transport to prove the theorem of Bubley and Dyer. As an application, we derived an upper bound for the mixing time of the exclusion process on a complete graph.

Notes: optimal_transport.pdf

May 24: We defined the transportation metric and discussed the main result of Bubley and Dyer on path couplings (section 14.2).

May 19: We estimated the bottleneck ratio of the Glauber dynamics for the coloring of star graphs. We concluded the lecture by discussing some basic results on distinguishing statistics (section 7.3).

Notes: lower_bounds.pdf

May 17: We focused on the derivation of lower bounds for mixing times and defined the bottleneck ratio of an ergodic chain (section 7.2).

May 10: We turned our attention to the notion of a strong stationary time and its use in deriving upper bounds for the mixing time of an ergodic chain (section 6.4).

Notes: strong_stationary_times.pdf

May 3 - May 5: In preparation for chapter 6, we introduced several measure theoretic concepts and discussed the definition of conditional expectation with respect to a σ-algebra (see also section A.2 in the appendix).

April 26: We used a grand coupling to estimate the mixing time of the Metropolis chain designed to sample proper colorings in graphs (section 5.4).

April 19 - April 21: We investigated the mixing times of various random walks on different geometries, including the n-cycle and the d-dimensional torus (section 5.3).

The third homework assignment has been posted.

April 12: We discussed the use of Markovian couplings in estimating the mixing time of an ergodic Markov chain.

April 7: We proved the convergence of irreducible and aperiodic Markov chains on finite state spaces (section 4.3).

April 5: We introduced the Doeblin condition and derived some basic results on geometric ergodicity.

Notes: MC_mixing.pdf

March 31: We discussed the coupling method and derived the fundamental relation between optimal couplings and the total variation distance (section 4.2).

March 29: We introduced the Metropolis algorithm and discussed its applicability to certain discrete optimization problems (section 3.2).

The second homework assignment has been posted.

Notes: MCMC.pdf

March 25: We defined the notion of a reversible Markov chain and turned our attention to Markov chain Monte Carlo methods.

March 17 - March 22: We derived the existence and uniqueness of stationary distributions for irreducible finite Markov chains (section 1.5).

March 15: We introduced the notions of irreducibility and aperiodicity for finite Markov chains.

The first homework assignment has been posted.

March 10: We discussed the computation of multi-step transition probabilities, and proved that every transition matrix on a finite state space has a random mapping representation (section 1.2).

March 8: We defined the notion of a Markov chain and focused on specific examples, such as nearest neighbor random walks and the gambler's ruin problem.

My lecture notes for the first chapter can be found here: finite_MCs.pdf