The scaling limit of the membrane model

2018/01/17 by Cipriani, Alessandra, Dan, Biltu, Hazra, Rajat Subhra · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.05663

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 Hölder 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 (2009) in d=1.

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