6 - First evidence of noise sensitivity of percolation

Summary

In this chapter, our goal is to collect some of the facts and theorems we have seen so far in order to conclude that percolation crossings are indeed noise sensitive. Recall from the “BKS” Theorem (Theorem 1.21) that it is enough for this purpose to prove that influences are “small” in the sense that Σ_kI_k(f_n)² goes to 0. If we want to use only what we have actually proved in this book, namely Proposition 5.6, then we need to demonstrate (5.6) in this proposition.

In the first section, we deal with a careful study of influences in the case of percolation crossings on the triangular lattice. Then, we treat the case of ℤ², where conformal invariance is not known. Finally, we speculate to what “extent” percolation is noise sensitive.

This whole chapter should be considered somewhat of a “pause” in our program, where we take the time to summarize what we have achieved so far in our understanding of the noise sensitivity of percolation, and what remains to be done if one wishes to obtain the exact “noise sensitivity exponent” as well as the existence of exceptional times for dynamical percolation.

Bounds on influences for crossing events in critical percolation on the triangular lattice

6.1.1 Setup

Fix a, b > 0, let us consider some rectangle [0, a·n]×[0, b·n], and let R_n be the set of hexagons in T that intersect [0, a·n]×[0, b·n]. Let f_n be the event that there is a left-to-right crossing event in R_n. (This is the same event as in Example 1.22 in Chapter 1, but with ℤ² replaced by T.) By the RSW Theorem 2.1, we know that {f_n} is nondegenerate. Conformal invariance tells us that E[f_n] = ℙ[f_n = 1] converges as n → ∞. This limit is given by the so-called Cardy's formula.