TY - JOUR

T1 - Random-field random surfaces

AU - Dario, Paul

AU - Harel, Matan

AU - Peled, Ron

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023

Y1 - 2023

N2 - 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 ∇ ϕ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions 1 ≤ d≤ 2 and localizes in dimensions d≥ 3. (ii) The surface delocalizes in dimensions 1 ≤ d≤ 4 and localizes in dimensions d≥ 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≥ 3. (ii) The surface delocalizes in dimensions d= 1 , 2. (iii) The surface localizes in dimensions d≥ 3 at low temperature and weak disorder strength. The behavior in dimensions d≥ 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.

AB - 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 ∇ ϕ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions 1 ≤ d≤ 2 and localizes in dimensions d≥ 3. (ii) The surface delocalizes in dimensions 1 ≤ d≤ 4 and localizes in dimensions d≥ 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≥ 3. (ii) The surface delocalizes in dimensions d= 1 , 2. (iii) The surface localizes in dimensions d≥ 3 at low temperature and weak disorder strength. The behavior in dimensions d≥ 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.

UR - http://www.scopus.com/inward/record.url?scp=85148621457&partnerID=8YFLogxK

U2 - 10.1007/s00440-022-01179-0

DO - 10.1007/s00440-022-01179-0

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AN - SCOPUS:85148621457

SN - 0178-8051

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

ER -