Noise sensitivity and the Fourier spectrum of critical percolation

Abstract: Consider an n-by-n box in the square grid Z^2. Delete each edge of the grid independently from each other with probability 1/2. By planar duality, the probability of still having a crossing between the left and right sides of the box is exactly 1/2. Assume now that we know about almost every edge if it is deleted or not, except for a very small proportion. Can we guess if there is a left-right crossing?

In joint work with Christophe Garban and Oded Schramm, we have found that the answer is a strong NO: "critical percolation is very noise sensitive". The reason is that "the Fourier-Walsh coefficients of the percolation crossing function are concentrated on high frequencies". The goal of this talk will be to explain what these notions and claims actually mean.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).