Random-field random surfaces

We study how the typical gradient and height of a random surface are modified by addition quenched disorder in form independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, following cases. It is shown that for real-valued random-field surfaces $$\nabla \phi $$ type with uniformly convex interaction potential: (i) delocalizes dimensions $$1\le d\le 2$$ localizes $$d\ge 3$$ . (ii) 4$$ 5$$ further integer-valued Gaussian free field: $$d=1,2$$ (iii) at low temperature weak strength. behavior high or strong left open. proofs rely on several tools: Explicit identities satisfied expectation surface, Efron–Stein concentration inequality, coupling argument Langevin dynamics (originally due Funaki Spohn (Comm Math Phys 185(1): 1-36, 1997) Nash–Aronson estimate.