  We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$
of discrete, probability measure-preserving equivalence relations, as the
sampling exponent of a generating random walk on the ambient relation. The
co-spectral radius is analogous to the spectral radius for random walks on
$G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more
general version of the 2-3 method we used in another work on the growth of
unimodular random rooted trees. We use this method to show that the walk growth
exists for an arbitrary unimodular random rooted graph of bounded degree. We
also investigate how the co-spectral radius behaves for hyperfinite relations,
and discuss new critical exponents for percolation that can be defined using
the co-spectral radius.
