Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees

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. The almost sure existence of the sampling exponent is already new for i.i.d. percolation clusters on countable groups. For the proof, we develop a general method called 2-3-method that is based on the mass-transport principle. As a byproduct, we show that the growth of a unimodular random rooted tree of bounded degree always exists, assuming its upper growth passes a critical threshold. This complements Timar's work who showed the possible nonexistence of growth below this threshold. We also 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 Property (T) and hyperfinite relations.Comment: 66 pages, 5 figures. Comments welcome