Maximilian Nitzschner and Alberto Chiarini: Phase transition for level-set percolation of the membrane model in dimensions d ≥ 5

Abstract: We consider level-set percolation for the Gaussian membrane model on \mathbb{Z}^d with d \geq 5 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 h_\ast, which we show to be positive in high dimensions. To characterize the percolation phase transition more precisely, two further natural critical levels h_{\ast\ast} and \overline{h} are introduced which bound h_\ast from above and below, respectively, and are shown to be finite in all dimensions. In the (strongly) subcritical phase h > h_{\ast\ast}, we provide bounds for the connectivity function of the level-set above h, and  in the (strongly) supercritical phase h < \overline{h} we characterize the geometry of the level-set above level h, by verifying conditions identified by Drewitz, Ráth and Sapozhnikov for general correlated percolation models.

In the first half of the talk, we focus on introducing the model along with the general context of level-set percolation, demonstrating differences and similarities to related models. In the second half of the talk, we present in detail novel decoupling inequalities for the membrane model, which are instrumental in the study of both the subcritical and supercritical phases of its level-sets, and discuss the positivity of h_\ast in high dimensions.


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