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The Dickman subordinator, renewal theorems, and disordered systems. (English) Zbl 07107385
Summary: We consider the so-called Dickman subordinator, whose Lévy measure has density \(\frac{1} {x}\) 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.

MSC:
60K05 Renewal theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60G51 Processes with independent increments; Lévy processes
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