A new proof of the sharpness of the percolation phase transition
The sharpness of the percolation phase transition, which is
a crucial and much cited element of the theory, was first proved
independently by Menshikov in 1986 and by Aizenman and Barsky in 1987.
On February 11, 2015, a remarkable new proof was posted on arXiv by
Duminil-Copin and Tassion. I will describe their proof, which
is completed in two pages, in its entirety.