# Week 3 - lecture 2

## TOP TOPIC

Date: 01/09/22

3.4. Birth and death chains
3.5. A markov chain from linear algebra
3.6. Counting and diameter lower bounds
3.7. Bottleneck Ratio

## Homework

1.* Let be simple random walk on the integers, with . Show that for all . Hint: Prove the inequality for all real , and use it to bound .

2.*  Let  be a b-ary tree of height , that has vertices.  What are the best upper and lower bounds you can prove for the mixing time of lazy SRW on ?

3. Let be the set of invertible matrices mod 2. Find the cardinality of . Hint: consider choosing the rows of the matrix one by one.

4.* Let be a subset of the group of permutations . consider the Markov chain on obtained by picking a permutation in uniformly at random (independently of previous ), and defining .

(a)  Write a lower bound for the mixing time of this chain in terms of and .

(b) Suppose that consists of transpositions. What is the  smallest possible size of for which this chain is irreducible?

(c) If consists of transpositions, can this chain be aperiodic?

(d) Suppose that consists of all transpositions, and we consider the lazy version of the chain above. Can you improve the lower bound from part (a) in this case?

## Reference

[1] Glasner, Shmuel.
Almost periodic sets and measures on the torus.
Israel J. Math. 32 (1979), no. 2-3, 161–-172.

[2] Berend, Daniel and Peres, Yuval.
Asymptotically dense dilations of sets on the circle.
J. London Math. Soc. (2) 47 (1993), no. 1, 1–-17.

[3] N. Alon and Y. Peres, "Uniform dilations'', Geometric and Functional Analysis, vol. 2, No. 1 (1992), 1–28.

[4] Nair, R. and Velani, S. L.
Glasner sets and polynomials in primes.
Proc. Amer. Math. Soc. 126 (1998), no. 10, 2835–-2840.