[Paper Review] Moderate deviations for the chemical distance in Bernoulli percolation

Olivier Garet, Régine Marchand|ArXiv.org|Jul 3, 2009

Stochastic processes and statistical mechanics19 references17 citations

TL;DR

This paper establishes moderate deviation estimates for the chemical distance in supercritical Bernoulli percolation on $\mathbb{Z}^d$, using concentration inequalities and subadditivity to bound fluctuations around the mean. It proves that the chemical distance $D(0,y)$ deviates from its mean $\mu(y)$ by at most $O(\sqrt{\|y\|_1} \log(1+\|y\|_1))$ with high probability, and provides exponential tail bounds for moderate deviations, improving understanding of the asymptotic shape of the percolation cluster.

ABSTRACT

In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.

Motivation & Objective

• To analyze the random fluctuations of the chemical distance in supercritical Bernoulli percolation on $\mathbb{Z}^d$.
• To establish moderate deviation estimates for the chemical distance $D(0,y)$ around its mean $\mu(y)$, beyond large deviations.
• To improve the understanding of the convergence to the asymptotic shape of the percolation cluster by quantifying typical fluctuations.
• To control the discrepancy between the expected chemical distance $\mathbb{E}[D^*(0,y)]$ and the deterministic norm $\mu(y)$.
• To develop a refined approximation scheme using $Q_x$-skeletons and renormalization to bound path lengths in the infinite cluster.

Proposed method

• The authors introduce a modified chemical distance $D^*(x,y) = D(x^*,y^*)$, where $x^*$ is the closest point in the infinite cluster $C_\infty$ to $x$, to handle disconnected points.
• They apply concentration inequalities from Boucheron, Lugosi, and Massart to control the deviation of $D^*(0,y)$ from its mean.
• Subadditivity and approximation theory of subadditive functions (inspired by Alexander) are used to bound the mean deviation $|\mathbb{E}[D^*(0,y)] - \mu(y)|$.
• A renormalization scheme is constructed using $Q_x$-paths and skeletons to decompose long paths into short and long increments, enabling variance and deviation control.
• The proof distinguishes between short increments (in $\Delta_x$) and long increments (in $D_x$), using separate bounds from Lemma 4.6 to control their counts.
• Exponential tail bounds are derived by combining moment estimates, concentration inequalities, and path decomposition, leading to bounds of the form $\mathbb{P}(|D^*(0,y) - \mathbb{E}[D^*(0,y)]| / \sqrt{\|y\|_1} > x) \leq A e^{-Bx}$ for $x \geq C(1 + \log \|y\|_1)$.

Experimental results

Research questions

• RQ1 How do the typical fluctuations of the chemical distance $D(0,y)$ behave around its mean $\mu(y)$ in supercritical Bernoulli percolation?
• RQ2 Can moderate deviation estimates be established for the chemical distance, with exponential tail decay, beyond the large deviation regime?
• RQ3 What is the quantitative control on the discrepancy between $\mathbb{E}[D^*(0,y)]$ and the asymptotic norm $\mu(y)$?
• RQ4 How can the path structure in the infinite cluster be approximated using $Q_x$-skeletons to bound the chemical distance?
• RQ5 What is the optimal scaling of the variance of $D^*(0,y)$, and how does it relate to $\|y\|_1$?

Key findings

• The expected deviation of the chemical distance from its mean satisfies $\mathbb{E}[|D(0,y) - \mu(y)| \mathbf{1}_{\{0\leftrightarrow y\}}}] \leq C_{\ref{concenD}} \sqrt{\|y\|_1} \log(1 + \|y\|_1)$.
• For $x \in [C_{\ref{equmoderter}}(1 + \log \|y\|_1), \|y\|_1^{1/2}]$, the moderate deviation probability satisfies $\mathbb{P}\left(\frac{|D(0,y) - \mu(y)|}{\sqrt{\|y\|_1}} > x, \; 0\leftrightarrow y\right) \leq A_{\ref{equmoderter}} e^{-B_{\ref{equmoderter}}x}$.
• The variance of the modified distance satisfies $\text{Var } D^*(0,y) \leq C_{\ref{equvar}} \|y\|_1 \log(1 + \|y\|_1)$.
• The mean of $D^*(0,y)$ satisfies $0 \leq \mathbb{E}[D^*(0,y)] - \mu(y) \leq C_{\ref{equetlesperance1}} \sqrt{\|y\|_1} \log(1 + \|y\|_1)$.
• The random ball $B^0(t) = \{x : D(0,x) \leq t\}$ is almost surely contained in a $\mu$-ball of radius $t + C_{\ref{equtfa}} \sqrt{t} \log t$, and contains the ball of radius $t - C_{\ref{equtfa}} \sqrt{t} \log t$.
• For $x \in [C_{\ref{equmoderbis}}(1 + \log \|y\|_1), \sqrt{\|y\|_1}]$, the tail bound $\mathbb{P}\left(\frac{|D^*(0,y) - \mu(y)|}{\sqrt{\|y\|_1}} > x\right) \leq A_{\ref{equmoderbis}} e^{-B_{\ref{equmoderbis}}x}$ holds with explicit constants.