[Paper Review] Lectures on noise sensitivity and percolation

Christophe Garban, Jeffrey E. Steif|arXiv (Cornell University)|Feb 28, 2011

Stochastic processes and statistical mechanics40 references14 citations

TL;DR

This paper establishes noise sensitivity and exceptional times in critical percolation on ℤ² using tools from Boolean function analysis and Fourier analysis on the hypercube. It proves the existence of exceptional times in dynamical percolation where infinite clusters almost surely appear, leveraging spectral analysis and the second moment method with quasi-multiplicativity and power-law bounds on crossing probabilities.

ABSTRACT

The present text provides the lecture notes for the course "noise sensitivity and percolation" given at the 2010 Clay Summer School in Buzios, Brazil.

Motivation & Objective

• To bridge statistical mechanics and Boolean function theory by interpreting percolation events as Boolean functions on the hypercube.
• To analyze noise sensitivity of percolation using Fourier analysis and spectral techniques.
• To prove the existence of exceptional times in dynamical percolation on ℤ² where infinite clusters appear almost surely.
• To extend results on sharp thresholds and influence to percolation models using KKL and Friedgut-Kalai theorems.
• To apply randomized algorithms and revealment theory to study critical exponents and spectral properties.

Proposed method

• Use discrete Fourier analysis on the hypercube to represent Boolean functions and analyze their energy spectrum.
• Apply hypercontractivity and the KKL theorem to bound influences and establish noise sensitivity.
• Employ the spectral sample technique to characterize high-frequency components of percolation events.
• Use the second moment method with quasi-multiplicativity to prove existence of exceptional times in dynamical percolation.
• Leverage power-law bounds on crossing probabilities (e.g., s(r) ≥ r^δ) and α₁(r)α₄(r) ≥ r^{ε₀−2} to control decay rates.
• Integrate spectral bounds with moment estimates to show integrability near zero, enabling the second moment argument.

Experimental results

Research questions

• RQ1 Do critical percolation configurations exhibit noise sensitivity under small independent perturbations of the underlying Bernoulli variables?
• RQ2 Can exceptional times exist in dynamical percolation on ℤ² where an infinite cluster appears almost surely despite criticality?
• RQ3 What is the role of spectral properties and Fourier concentration in determining noise sensitivity of percolation events?
• RQ4 How do influence and revealment in Boolean functions relate to critical exponents and phase transitions in percolation?
• RQ5 Can the second moment method be applied to dynamical percolation using spectral and crossing probability estimates?

Key findings

• The paper proves that there exist almost surely exceptional times in dynamical percolation on ℤ² at p_c = 1/2 where an infinite cluster exists.
• Noise sensitivity of percolation is established via spectral analysis, showing that the spectral sample concentrates near the upper bound.
• The second moment method succeeds due to integrability of the correlation decay, ensured by α₁(s(1/t))^{-1} ≤ O(1) t^{η} for some η > 0.
• Quasi-multiplicativity and power-law bounds on crossing probabilities (s(r) ≥ r^δ) are essential for controlling the decay of correlations.
• The existence of exceptional times is shown to hold under general conditions, including on spherically symmetric trees with ∑ 1/(np^{-n}T_n) < ∞.
• The critical value p_c = 1/2 for ℤ² and the torus 𝕋 is confirmed using sharp threshold theorems and influence analysis.