Abstract:
A random walk on a finitely generated group G is a sequence of the form W_n=X_1...X_n, where X_1,X_2,... are i.i.d. random variables distributed uniformly on some (finite) generating set S of G. Letting |·| denote the word metric with respect to S, we study the asymptotic behavior of the distance of the random walk W_n. The speed of the random walk E[|W_n|], i.e. its expected distance from its starting position, has been studied over the last years. In particular, works by Erschler, Amir-Virág and Brieussel-Zheng realized many functions as speed functions of random walks on groups.

In a joint work with Gideon Amir, we study the construction of Brieussel-Zheng, showing that random walks on these groups satisfy a law of iterated logarithm. This gives, for any 1/2\le β\le 1, an example of a group with speed of order n^β that satisfies
\[
    0 < \limsup_{n\to\infty} |W_n|/(n^β(\log\log n)^{1-β}) < \infty
\]
and
\[
    0 < \liminf_{n\to\infty} (|W_n|(\log\log n)^{1-β})/n^β < \infty
\]
almost surely. Relevant background will be covered in the talk.