Rigidity of harmonic functions on the supercritical percolation cluster

Abstract

We use ideas from quantitative homogenization to show that nonconstant harmonic functions on the percolation cluster cannot satisfy certain structural constraints, for example, a Lipschitz bound. These unique-continuation-type results are false on the full lattice and hence the disorder is utilized in an essential way.

Figures (25)

• Figure 1: The effect of changing the value of an edge. On the left is the harmonic (barycentric) embedding of the cluster for $\mathfrak{p} = 0.8$ on a cube of side length $N=50$ superimposed with the embedding after one edge is re-sampled. The original embedding is in blue and the re-sampled embedding is shown in green. On the right is a zoomed in piece of the re-sampled embedding.
• Figure 2: An example of the decomposition in the proof of Lemma \ref{['lemma:jordan-curves-theorem']}. The curves $\alpha_1, \alpha_2, \alpha_3$ are denoted by blue, red, and navy arcs respectively. The gray curves are the infinite paths $\gamma_i$. The concatenated curve $\beta_3$ defined in \ref{['eq:concatenated-curve']} is shown in black on the right together with the complementary connected components $C_{3}^{\pm}$.
• Figure 3: Subsets of the box of radius $N$, $Q_N$, as defined in \ref{['eq:box-decomp']} and used in the proof of Proposition \ref{['prop:not-lipschitz']}.
• Figure 4: Partitions of $Q_N$ used in the proof of Proposition \ref{['prop:not-lipschitz']}. Some of the small cubes and medium cubes which are adjacent to $\hbox{center}(Q_N)$ are shown. All small boxes other than the blue ones have had associated horizontal edges removed according to the algorithm described at the beginning of Step 2.
• Figure 5: The projection on $\hbox{center}(Q_N)$ of a medium cube (whose size is $N^{1/(10d)}$) partitioned into small cubes (whose sizes are $N^{1/(10d)^2}$). A boundary layer of blue cubes is depicted in blue. The horizontal edges of the blue cubes are not altered.

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Theorem 2.4: Theorem 1.2 in armstrong-dario-2018
• Theorem 2.5: (1.22) in armstrong-dario-2018 and Proposition 2.12 in dario-gu-green