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We consider the so-called Dickman subordinator, whose Levy measure hasdensity 1/x restricted to the interval (0,1). The marginal density of thisprocess, known as the Dickman function, appears in many areas of mathematics,from number theory to combinatorics. In this paper, we study renewal processesin the domain of attraction of the Dickman subordinator, for which we provelocal renewal theorems. We then present applications to marginally relevantdisordered systems, such as pinning and directed polymer models, and provesharp second moment estimates on their partition functions.

The Dickman subordinator, renewal theorems, and disordered systems