[Paper Review] Noise sensitivity in bootstrap percolation

Zsolt Bartha, Gábor Pete|arXiv (Cornell University)|Sep 28, 2015

Stochastic processes and statistical mechanics27 references3 citations

TL;DR

This paper investigates noise sensitivity in $k$-neighbour bootstrap percolation on finite graphs, showing that complete occupation on the $d$-dimensional torus $[n]^d$ at critical density is noise sensitive for $k=2$, $d\geq2$, while on $d$-regular random graphs $G_{n,d}$, it is noise insensitive for $2\leq k\leq d-2$. The result hinges on the critical density's behavior: $p_c \to 0$ in the torus (noise sensitive), $p_c > 0$ in random graphs (insensitive).

ABSTRACT

Answering questions of Itai Benjamini, we show that the event of complete occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d, for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for 2\leq k\leq d-2, it is insensitive. Many open problems remain.

Motivation & Objective

• To determine whether the event of complete occupation in $k$-neighbour bootstrap percolation is noise sensitive or insensitive at the critical initial density $p_c(n)$ on finite graphs.
• To resolve a question posed by Itai Benjamini regarding noise sensitivity in bootstrap percolation on $[n]^d$ and $G_{n,d}$.
• To investigate the relationship between the asymptotic behavior of $p_c(G_n,k)$ and the noise sensitivity of the complete occupation event.
• To establish a contrast between the noise sensitivity of bootstrap percolation on Euclidean-like lattices and random regular graphs, despite both having monotone dynamics.

Proposed method

• Use discrete Fourier analysis and noise operator techniques to analyze correlation decay between the initial configuration and a noisy version where each bit is resampled independently with probability $\epsilon$.
• Apply Russo's formula and pivotality estimates to bound the influence of individual vertices on the complete occupation event, showing $\mathbb{I}_x = \mathbb{P}_p(x \text{ is pivotal}) = O(\log^{3+o(1)}n / n^2)$ for $d=2$.
• Use the noise stability bound $\mathcal{S}_\varepsilon(f) \leq (6e+1) \cdot \mathcal{W}(f)^{\alpha(\varepsilon) \cdot \varepsilon}$, where $\mathcal{W}(f)$ is the total influence and $\alpha(\varepsilon)$ depends on the logarithmic scale of $p_c(n)$.
• Leverage the known asymptotic behavior of $p_c([n]^d,2) = \Theta(1 / \log^{d-1}n)$ and $p_c(G_{n,d},k) \to p_c(\mathbb{T}_d,k) > 0$ to compare noise sensitivity across graph types.
• Use coupling arguments between the torus $\mathbb{T}_n^d$ and the box $[n]^d$, showing that $p_c([n]^d) = p_c(\mathbb{T}_n^d) + o(1/n^{1/3})$, and that noise stability decays uniformly.
• Establish that for $\epsilon_n \gg \log\log n / \log n$, the correlation $\mathrm{Corr}(f^\mathcal{C}_n(\omega), f^\mathcal{C}_n(\omega^{\epsilon_n})) \to 0$ on $[n]^d$, proving noise sensitivity.

Experimental results

Research questions

• RQ1 Is the complete occupation event in 2-neighbour bootstrap percolation on $[n]^d$ noise sensitive at the critical density $p_c(n)$?
• RQ2 Is the complete occupation event in $k$-neighbour bootstrap percolation on $d$-regular random graphs $G_{n,d}$ noise sensitive or insensitive for $2 \leq k \leq d-2$?
• RQ3 Does the noise sensitivity of the complete occupation event correlate with whether $p_c(G_n,k) \to 0$ or $\liminf p_c(G_n,k) > 0$?
• RQ4 Can the noise sensitivity of bootstrap percolation be characterized via the influence of individual vertices and the structure of the critical window?
• RQ5 How does the behavior of the noise operator $N_p^\epsilon$ on the indicator function of the complete occupation event relate to the decay of correlation under noise?

Key findings

• For $d \geq 2$, the complete occupation event in 2-neighbour bootstrap percolation on $[n]^d$ is noise sensitive: for any $\epsilon_n \gg \log\log n / \log n$, the correlation between the original and noisy configurations tends to zero as $n \to \infty$.
• On $d$-regular random graphs $G_{n,d}$, the complete occupation event is noise insensitive for $2 \leq k \leq d-2$, as the correlation remains bounded away from zero even under small noise.
• The pivotal influence $\mathbb{I}_x$ of each vertex $x$ on the complete occupation event satisfies $\mathbb{I}_x = O(\log^{3+o(1)}n / n^2)$ for $d=2$, leading to total influence $\mathcal{W}(\mathcal{C}_{\mathbb{T}_n^2}) = O(\log^{5+o(1)}n / n^2)$.
• The noise stability decays as $O\left(\left(\frac{\log^{5+o(1)}n}{n^2}\right)^{\varepsilon \cdot \frac{1}{\log\log n}}\right)$, and this tends to zero when $\epsilon_n \gg \log\log n / \log n$, confirming noise sensitivity on the torus.
• The critical density $p_c([n]^d,2)$ satisfies $p_c([n]^d,2) = p_c(\mathbb{T}_n^d,2) + o(1/n^{1/3})$, allowing transfer of noise sensitivity results from the torus to the box.
• The result suggests a general dichotomy: noise sensitivity occurs when $p_c \to 0$, insensitivity when $p_c > 0$, though a full explanation remains open.