## Sharp Thresholds and Extremal Combinatorics

Dor Minzer

Member, Institute for Advanced Study

March 17, 2020

Consider the p-biased distribution over 0,1n, in which each coordinate independently is sampled according to a p-biased bit. A sharp-threshold result studies the behavior of Boolean functions over the hypercube under different p-biased measures, and in particular whether the function experiences a phase transition between two, close p's. While the theory of sharp-thresholds is well understood for p's that are bounded away from 0 and 1, it is much less so for values of p that are close to 0 or 1.