Abstract: We consider level-set percolation for the Gaussian membrane model on with and establish that as varies, a non-trivial percolation phase transition for the level-set above level occurs at some finite critical level , which we show to be positive in high dimensions. To characterize the percolation phase transition more precisely, two further natural critical levels and are introduced which bound from above and below, respectively, and are shown to be finite in all dimensions. In the (strongly) subcritical phase , we provide bounds for the connectivity function of the level-set above , and in the (strongly) supercritical phase we characterize the geometry of the level-set above level , 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 in high dimensions.
Based on https://arxiv.org/abs/2112.09116.