Summary: Non-backtracking random walks mix faster
Sasha Sodin §
October 18, 2006
We compute the mixing rate of a non-backtracking random walk on a regular expander.
Using some properties of Chebyshev polynomials of the second kind, we show that this rate
may be up to twice as fast as the mixing rate of the simple random walk. The closer the
expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is.
As an application, we show that if G is a high-girth regular expander on n vertices, then
a typical non-backtracking random walk of length n on G does not visit a vertex more than
(1 + o(1)) log n
log log n times, and this result is tight. In this sense, the multi-set of visited vertices is
analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random
walk on G, which almost surely visits some vertex (log n) times.
1.1 Background and definitions
Let G = (V, E) be an undirected graph. A random walk of length k on G, from some given vertex