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Co-spectral radius for countable equivalence relations
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 12 / December 2024
- Published online by Cambridge University Press:
- 10 May 2024, pp. 3385-3427
- Print publication:
- December 2024
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- Article
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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.
