Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$

Autor: Chiarini, Alberto, Nitzschner, Maximilian
Rok vydání: 2021
Předmět:
Zdroj: J. Stat. Phys., 190, 59, 2023
Druh dokumentu: Working Paper
DOI: 10.1007/s10955-023-03072-z
Popis: We consider level-set percolation for the Gaussian membrane model on $\mathbb{Z}^d$, with $d \geq 5$, and establish that as $h \in \mathbb{R}$ 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. Along $h_\ast$, two further natural critical levels $h_{\ast\ast}$ and $\overline{h}$ are introduced, and we establish that $-\infty <\overline{h} \leq h_\ast \leq h_{\ast\ast} < \infty$, in all dimensions. For $h > h_{\ast\ast}$, we find that the connectivity function of the level-set above $h$ admits stretched exponential decay, whereas for $h < \overline{h}$, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, R\'ath and Sapozhnikov, see arXiv:1212.2885, for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model.
Comment: 29 pages, 1 figure, to appear in Journal of Statistical Physics
Databáze: arXiv
Externí odkaz:
Nepřihlášeným uživatelům se plný text nezobrazuje