Open Access
2019 The Dickman subordinator, renewal theorems, and disordered systems
Francesco Caravenna, Rongfeng Sun, Nikos Zygouras
Electron. J. Probab. 24: 1-40 (2019). DOI: 10.1214/19-EJP353


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.


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Francesco Caravenna. Rongfeng Sun. Nikos Zygouras. "The Dickman subordinator, renewal theorems, and disordered systems." Electron. J. Probab. 24 1 - 40, 2019.


Received: 22 October 2018; Accepted: 10 August 2019; Published: 2019
First available in Project Euclid: 18 September 2019

zbMATH: 07107385
MathSciNet: MR4017119
Digital Object Identifier: 10.1214/19-EJP353

Primary: 60K05
Secondary: 60G51 , 82B44

Keywords: Dickman function , Dickman subordinator , directed polymer model , Disordered system , Levy process , pinning model , Renewal process , Renewal theorem , Stable process

Vol.24 • 2019
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