Random-field random surfaces

TL;DR

This paper investigates how quenched disorder affects the localization and delocalization of random surfaces across various dimensions, providing sharp quantitative estimates for different models and disorder strengths.

Contribution

It offers new rigorous results on the dimensional thresholds for localization and delocalization in disordered random surface models, including both real-valued and integer-valued cases.

Findings

Gradient delocalizes in low dimensions, localizes in high dimensions.

Surface delocalizes in dimensions up to 4, localizes in 5 and above.

Results depend on the model type and disorder strength.

Abstract

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued disordered random surfaces of the $\nabla \phi$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes in dimensions $d\ge 5$. It is further shown that for the integer-valued disordered Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at weak disorder strength. The behavior in dimensions $d\ge 3$ at strong disorder is left open. The proofs rely on several tools: explicit identities satisfied by the expectation of the random surface, the Efron--Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn) and the Nash--Aronson estimate.