Influence of spatial correlation for directed polymers

TL;DR

This paper investigates how strong spatial correlation in the environment affects the behavior of directed polymers, revealing conditions for superdiffusivity and strong disorder across dimensions.

Contribution

It introduces new results on superdiffusivity and disorder in directed polymers with correlated environments, extending previous models and providing novel analytical approaches.

Findings

Superdiffusivity holds at all temperatures for dimensions d≥3.

Strong disorder persists under certain conditions in correlated environments.

New methods are developed for analyzing Brownian pinning models with power-law decay.

Abstract

In this paper, we study a model of a Brownian polymer in $\mathbb {R}_+\times \mathbb {R}^d$, introduced by Rovira and Tindel [J. Funct. Anal. 222 (2005) 178--201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when $d\ge3$ and provide a novel approach to Petermann's superdiffusivity result in dimension one [Superdiffusivity of directed polymers in random environment (2000) Ph.D. thesis]. We also derive results for a Brownian model of pinning in a nonrandom potential with power-law decay at infinity.