Critical continuum random polymer measures M_r are constructed on diamond fractals, and intersections of two independent paths are shown to have Hausdorff dimension zero with log-Hausdorff exponent 1.
The Dickman subordinator, renewal theorems, and disordered systems
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abstract
We consider the so-called Dickman subordinator, whose Levy measure has density 1/x restricted to the interval (0,1). The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.
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Continuum models of directed polymers on disordered diamond fractals in the critical case
Critical continuum random polymer measures M_r are constructed on diamond fractals, and intersections of two independent paths are shown to have Hausdorff dimension zero with log-Hausdorff exponent 1.