Electronic Journal of Probability

The Dickman subordinator, renewal theorems, and disordered systems

Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1568793795

Digital Object Identifier
doi:10.1214/19-EJP353

Mathematical Reviews number (MathSciNet)
MR4017119

Zentralblatt MATH identifier
07107385

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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