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One of the most popular models for quantitatively understanding the emergenceof drug resistance both in bacterial colonies and in malignant tumors wasintroduced long ago by Luria and Delbr\"uck. Here, individual resistant mutantsemerge randomly during the birth events of an exponentially growing sensitivepopulation. A most interesting limit of this process occurs when the populationsize $N$ is large and mutation rates are low, but not necessarily smallcompared to $1/N$. Here we provide a scaling solution valid in this limit,making contact with the theory of Levy $\alpha$-stable distributions, inparticular one discussed long ago by Landau. One consequence of thisassociation is that moments of the distribution are highly misleading as far ascharacterizing typical behavior. A key insight that enables our solution isthat working in the fixed population size ensemble is not the same as workingin a fixed time ensemble. Some of our results have been presented previously inshortened form.

Scaling solution in the large population limit of the general asymmetric stochastic Luria-Delbr\"uck evolution process