1.* Let be simple random walk on the hypercube , and let denote the Hamming weight (i.e., the number of ones) of . Then is also a Markov chain, with transition probabilities and for . Read about projected (or lumped) chains on pages 23-25 of the textbook, and calculate for . Hint: it suffices to calculate .

2. Show that the mixing time of lazy simple random walk on the discrete torus (see section 5.3.3 and Theorem 5.6 of the book) is .

3*. Consider the following Markov chain on the integers: Fix . Let for all integers . Let . For which is the hitting time finite almost surely, and what is its expectation ?

A key issue is determining when the expectation is finite. It is OK to use the strong law of large numbers.

4. Given an irreducible aperiodic transition matrix , show that for any two states , there is a coupling of the chains started from respectively, such that the expected coalescence time of these chains is at most .

5. Solve exercises 4.3 and 4.4 in the book, which can be found at https://www.yuval-peres-books.