Consider critical Bernoulli percolation on the planar triangular lattice, where each vertex is colored black or white with probability 1/2, independently from the other vertices. In 1999, [Benjamini-Kalai-Schramm] proved that this model is noise sensitive in the sense that, if we resample a small proportion of vertices, then we obtain a quasi-independent configuration (from the point of view of macroscopic percolation events). [Schramm-Steif] and [Garban-Pete-Schramm] then gave a precise description of this phenomenon.
All of these works are based on Fourier tools, that are very rich and beautiful, but are not very robust in the sense that it is hard to extend them to non-i.i.d. contexts. With Vincent Tassion, we have developped non-Fourier tools inspired by Kesten's study of near-critical percolation, and we:
- have given a new proof of the [Schramm-Steif] and [Garban-Pete-Schramm] theorems;
- have extended the results to the Ising model at high temperature under Glauber dynamics.
One of our main goals is to prove noise sensitivity for FK percolation; if we have time, we are also going to explain some ideas and partial results for this open problem.