Phase transition for level-set percolation of the membrane model

Abstract: We consider level-set percolation for the Gaussian membrane model on the integer lattice in dimensions five and higher, and establish that as h varies, a non-trivial percolation phase transition for the level-set above level h occurs at some finite critical level, which we show to be positive in high dimensions. Moreover, we demonstrate the existence of a strongly subcritical phase, in which we provide bounds for the connectivity function of the level-set above h, and a strongly supercritical phase, in which we characterize the geometry of the level-set above level h. As a main tool, we present novel decoupling inequalities for the membrane model, which are instrumental in the study of both the subcritical and supercritical phases of its level-sets. This talk is based on joint work with Alberto Chiarini.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

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