Exclusion Sensitivity of Boolean Functions
Recently the study of noise sensitivity and noise stability of Boolean functions has received
considerable attention. The purpose of this paper is to extend these notions in a natural way
to a different class of perturbations, namely those arising from running the symmetric
exclusion process for a short amount of time. In this study, the case of monotone Boolean
functions will turn out to be of particular interest. We show that for this class of functions,
ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion
process are equivalent. We also show this equivalence with respect to stability.
After obtaining these fairly general results, we study ``exclusion sensitivity'' of critical
percolation in more detail with respect to medium-range dynamics. The exclusion dynamics,
due to its conservative nature, is more physical than the classical i.i.d. dynamics. Interestingly,
we will see that in order to obtain a precise understanding of the exclusion sensitivity
of percolation, we will need to describe how typical spectral sets of percolation diffuse under
the underlying exclusion process.