The Dickman subordinator, renewal theorems, and disordered systems

TL;DR

This paper studies the Dickman subordinator and its renewal properties, applying these results to disordered systems like pinning and directed polymers, providing new renewal theorems and moment estimates.

Contribution

It introduces local renewal theorems for processes in the domain of attraction of the Dickman subordinator and applies them to disordered systems, offering novel analytical tools.

Findings

Proved local renewal theorems for the Dickman subordinator.

Derived sharp second moment estimates for partition functions.

Applied renewal results to disordered systems like pinning and polymers.

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.