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Abstract: Unimodular random graphs are probabilistic objects arising, for example, as the limits of sequences of finite graphs or as the connected components of a percolation on a transitive graph. In general, a unimodular random graph might fail to have an exponential growth rate but for unimodular random trees there is a curious dichotomy where the growth can be shown to exist once the “upper growth” passes certain threshold. Based on a joint work with Miklos Abert and Ben Hayes.