Random-field random surfaces

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 random-field 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 $$d\ge 3$$ . (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 random-field Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $$d=1,2$$ and localizes in dimensions $$d\ge 3$$ . (ii) The surface delocalizes in dimensions $$d=1,2$$ . (iii) The surface localizes in dimensions $$d\ge 3$$ at low temperature and weak disorder strength. The behavior in dimensions $$d\ge 3$$ at high temperature or 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 (Comm Math Phys 185(1): 1-36, 1997) and the Nash–Aronson estimate.