## Lecture 8: Confining a random walk on an expander is hard

Let be an -spectral expander, be a subset of vertices of of size . Suppose we uniformly choose a random vertex of and walk randomly for steps.

**Theorem 1 (Confining a random walk is hard). **Let be the event that the walk is confined within the entire time.Then, Prob. In particular, if then this confinement probability is exponentially small.

*Proof.* Let be the “*projection into matrix*“, i.e. and for all other . Noting that is idempotent, it is not difficult to see that

Prob

Thus, to bound Prob we need to know how much the matrix *shrinks *a vector after each multiplication. The same trick we did in the last lecture gives, for any non-zero vector

Now, consider any non-zero vector . Let . Then, . First, express as a linear combination of the orthonormal eigenvectors of :

Let and . Then,

The usual trick gives

By Cauchy-Schwarz, , we conclude that

Consequently, each time we apply to a vector we reduce its length by a ratio of at least . There are applications, and the initial vector has length . The theorem follows.

**Theorem 2 (Saving random bits for RP).** A language is in the class RP if there exists a polynomial time randomized algorithm satisfying the following conditions:

- Prob accepts
- Prob accepts

Suppose uses random bits. To reduce the error probability to , the easy way is to run the algorithm times and accept the input if the algorithm accepts it at least once. However, this approach requires random bits. By imposing a (strongly explicit) -spectral expander on the space of random strings of length , we can obtain the same effect with only random bits. Here, we require . A strongly explicit expander is an expander which, given a vertex, there’s a poly-time algorithm computing the neighboring vertices.

leave a comment