For
$G$ a finitely
generated group and
$g\in G$,
we say
$g$ is detected by
a normal subgroup
$N\u25c3G$
if
$g\notin N$. The
depth
${D}_{G}\left(g\right)$
of
$g$
is the lowest index of a normal, finite index subgroup
$N$ that detects
$g$. In this paper we study
the expected depth,
$\mathbb{E}\left[{D}_{G}\left({X}_{n}\right)\right]$,
where
${X}_{n}$ is a
random walk on
$G$.
We give several criteria that imply that
$$\mathbb{E}\left[{D}_{G}\left({X}_{n}\right)\right]\underset{n\to \infty}{\overset{}{\to}}2+\sum _{k\ge 2}\frac{1}{\left[G:{\Lambda}_{k}\right]}\phantom{\rule{0.3em}{0ex}},$$
where
${\Lambda}_{k}$
is the intersection of all normal subgroups of index at most
$k$.
In particular, the equality holds in the class of all nilpotent groups
and in the class of all linear groups satisfying Kazhdan’s property
$\left(T\right)$. We
explain how the righthand side above appears as a natural limit and also give an
example where the convergence does not hold.
