Electronic Journal of Probability

The Dickman subordinator, renewal theorems, and disordered systems

Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 101, 40 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60G51: Processes with independent increments; Lévy processes

Dickman subordinator Dickman function renewal process Levy process renewal theorem stable process disordered system pinning model directed polymer model

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Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos. The Dickman subordinator, renewal theorems, and disordered systems. Electron. J. Probab. 24 (2019), paper no. 101, 40 pp. doi:10.1214/19-EJP353. https://projecteuclid.org/euclid.ejp/1568793795

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