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

Alberto Chiarini, Maximilian Nitzschner

Abstract

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, Ráth 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.

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

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. Along , two further natural critical levels and are introduced, and we establish that , in all dimensions. For , we find that the connectivity function of the level-set above admits stretched exponential decay, whereas for , 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, Ráth 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.
Paper Structure (7 sections, 12 theorems, 139 equations, 1 figure)

This paper contains 7 sections, 12 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and useful results
  3. Decoupling inequality
  4. The subcritical phase
  5. The supercritical phase
  6. Positivity of in high dimensions
  7. Improved bounds on $G - G_N$

Key Result

Lemma 2.1

Let $I_k(\cdot)$ be the modified Bessel function of order $k\in \mathbb N$, and set $I_{-k} \stackrel{\mathrm{def}}{=} I_k$ for $k\in \mathbb N$. Then, for all $d\geq 5$ and for all $x = (x_1,\ldots,x_d)\in \mathbb Z^d$, As a consequence

Figures (1)

  • Figure 1: An illustration of the set-up in which a bound on the probability of $H_\varepsilon^c$ can be obtained.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 19 more