The truncated 0-stable subordinator, renewal theorems, and disordered systems

Francesco Caravenna, Rongfeng Sun, Nikos Zygouras

Published 2018-05-03Version 1

We introduce the subordinator, which we call "truncated 0-stable", whose Levy measure has density 1/x restricted to the interval (0,1). This process emerges naturally in the study of marginally relevant disordered systems, such as pinning and directed polymer models. We show that the truncated 0-stable subordinator admits an explicit marginal density and we study renewal processes in its domain of attraction, for which we prove sharp local renewal theorems. As an application, we derive sharp estimates on the second moment of the partition functions of pinning and directed polymer models.