Random-field random surfaces

Autor: Paul Dario, Matan Harel, Ron Peled
Rok vydání: 2023
Předmět:
Zdroj: Probability Theory and Related Fields. 186:91-158
ISSN: 1432-2064
0178-8051
Popis: 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.
Comment: Fixes a mistake in the appendix: the inequality (A.10) of arXiv v4 was incorrect, this is corrected in this version with an additional half a page. The result of Proposition 3.3 is still correct. ArXiv v4 is the version accepted for publication; 50 pages
Databáze: OpenAIRE
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