Phase transition for level-set percolation of the membrane model

Nitzschner Maximilian, The Hong Kong University of Science and Technology
Chiarini Alberto, Università degli Studi di Padova

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.

Area: IS12 - Random surfaces (Alessandra Cipriani)

Keywords: Membrane model, level-set percolation, decoupling inequalities

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