Noise sensitivity in percolation
Robert Morris (IMPA)

Suppose that in a close election, a small (random) proportion of the votes are accidentally miscounted; is this random `noise' likely to change the outcome of the election? It turns out that the answer to this question depends in interesting ways on the rule (i.e., the Boolean function f) by which the winner is selected. To take three simple examples, the answer is ``no'' if the function f is `majority' or `dictator', but ``yes'' if it is `parity'.
The systematic study of this problem was begun in 1999 by Benjamini, Kalai and Schramm, who gave a sufficient condition (based on the discrete Fourier coefficients of f) for the answer to be ``yes'', and used this result to prove that bond percolation on Z² is noise sensitive at criticality. More precisely, suppose that we perform critical (i.e., p = 1/2) bond percolation on Z², observe that there is a horizontal crossing of a particular n x n square, and then re-randomize each edge with probability epsilon > 0. Then the probability of having a horizontal crossing in the new configuration is close to 1/2.
In this talk we consider the corresponding question for continuum percolation, and in particular for the Poisson Boolean model (also known as the Gilbert disc model). Let eta be a Poisson process of density lambda in the plane, and connect two points of eta by an edge if they are at distance at most 1. We prove that, at criticality, the event that there is a crossing of an n x n square is noise sensitive. The proof is based on two extremely general tools: a version of the BKS Theorem for product measure, and a new extremal result on hypergraphs.
This is joint work with Daniel Ahlberg, Erik Broman and Simon Griffiths.