Abstract
On the integer lattice, we consider the discrete membrane model, a random interface in which the field has Laplacian interaction. We prove that, under appropriate rescaling, the discrete membrane model converges to the continuum membrane model in d ≥ 2. Namely, it is shown that the scaling limit in d = 2, 3 is a Holder continuous random field, while in d ≥ 4 the membrane model converges to a random distribution. As a by-product of the proof in d = 2, 3, we obtain the scaling limit of the maximum. This work complements the analogous results of Caravenna and Deuschel (Ann. Probab. 37 (2009) 903-945) in d = 1.
Original language | English |
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Pages (from-to) | 3963-4001 |
Number of pages | 39 |
Journal | Annals of Probability |
Volume | 47 |
Issue number | 6 |
DOIs | |
Publication status | Published - 2019 |
Keywords
- Continuum membrane model
- Green's function
- Membrane model
- Random interface
- Scaling limit