We study random string-duplication systems, which we call P\'olya string
models. These are motivated by DNA storage in living organisms, and certain
random mutation processes that affect their genome. Unlike previous works that
study the combinatorial capacity of string-duplication systems, or various
string statistics, this work provides exact capacity or bounds on it, for
several probabilistic models. In particular, we study the capacity of noisy
string-duplication systems, including the tandem-duplication, end-duplication,
and interspersed-duplication systems. Interesting connections are drawn between
some systems and the signature of random permutations, as well as to the beta
distribution common in population genetics
