For sparse locally convergent graph sequences, the exponential fast/slow threshold is bounded below by the local limit's survival threshold, and stretched-exponential extinction is possible below it.
Super-exponential extinction time of the contact process on random geometric graphs
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Phase transitions for contact processes on sparse random graphs via metastability and local limits
For sparse locally convergent graph sequences, the exponential fast/slow threshold is bounded below by the local limit's survival threshold, and stretched-exponential extinction is possible below it.