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We study the persistent homology of random \v{C}ech complexes. Generalizing amethod of Penrose for studying random geometric graphs, we first describe anappropriate theoretical framework in which we can state and address our mainquestions. Then we define the kth persistent Betti number of a random \v{C}echcomplex and determine its asymptotic order in the subcritical regime. Thisextends a result of Kahle on the asymptotic order of the ordinary kth Bettinumber of such complexes to the persistent setting.

Persistent Betti numbers of random \v{C}ech complexes