We analyze a crossover between ergodic and non-ergodic regimes in an interacting spin chain with a dilute density of impurities, defined as spins with a strong local potential. The dilute limit allows us to greatly suppress finite size effects and understand the mechanism of delocalization of these impurities in the thermodynamic limit. In particular, we show that at any finite impurity potential, impurities can always relax by exchanging energy with the rest of the chain. The relaxation rate only weakly depends on the impurity density and decays exponentially, up to logarithmic corrections, with the impurity potential. At finite impurity density the system will appear to be localized over a wide range of system sizes. However, this is a transient effect. We discuss the implications of this model on disordered systems and argue that it provides a generic mechanism for melting of the many body localized phase in the thermodynamic limit through weakly coupled conducting clusters.
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