Mathematics Colloquia and Seminars

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Let each site of the triangular lattice (or edge of the \Z^2 lattice) have an independent Poisson clock switching between open and closed. So, at any given moment, the configuration is just critical percolation. In particular, the probability of a left-right open crossing in an n*n box is roughly 1/2, and, on the infinite lattice, almost surely there are only finite open clusters. In the box, how long do we have to wait before we lose essentially all information about having a left-right open crossing? In the infinite lattice, are there random exceptional times when there are infinite clusters? In joint work with Christophe Garban and Oded Schramm, we gave quite complete answers: exceptional times do exist on both lattices, and the Hausdorff dimension of their set is computed to be 31/36 for the triangular lattice. The indicator function of a percolation crossing event is a function on the hypercube {-1,+1}^{sites or edges}, and thus it has a Fourier-Walsh expansion. Our proofs are based on giving sharp concentration results for the ``weight'' of the Fourier coefficients at different frequencies.