Noise sensitivity and FK-type representations for Gaussian and stable processes

Doctoral thesis, 2019

This thesis contains four papers on probability theory.

Paper A concerns the question of whether the exclusion sensitivity and exclusion stability of a sequence of Boolean functions are monotone with respect to adding edges to the underlying sequence of graphs.

In paper B, we use the tools developed in Paper A to give an elementary proof of the behaviour of the mixing time of a random interchange process on a complete graph.

In Paper C we discuss the relationship between the noise sensitivity, noise stability and volatility of sequences of Boolean functions. In particular, we show that the set of volatile such sequences is dense in the set of all sequences of Boolean functions. Moreover, we construct a noise stable and volatile sequence of Boolean functions which is not o(1)-close to any non-volatile sequence of Boolean functions.

Finally, in Paper D, we investigate which threshold Gaussian and threshold stable random vectors have color representations. We discuss this from many different perspectives, and results include formulae for the dimension of the kernel of the associated linear operator, geometric conditions on the Gaussian vectors whose threshold have color representations and explicit examples of stable vectors with phase transitions at any stability index for the corresponding threshold process to have a color representation for large thresholds .