A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of `topological events' of
smooth Gaussian fields on manifolds; these are events that depend only on the
topology of the level sets of the field, for example (i) crossing events for
level or excursion sets, (ii) events measurable with respect to the number of
connected components of level or excursion sets of a given diffeomorphism
class, and (iii) persistence events. As an application of the covariance
formula, we derive strong mixing bounds for topological events, as well as
lower concentration inequalities for additive topological functionals (e.g. the
number of connected components) of the level sets that satisfy a law of large
numbers. The covariance formula also gives an alternate justification of the
Harris criterion, which conjecturally describes the boundary of the percolation
university class for level sets of stationary Gaussian fields. Our work is
inspired by a recent paper by Rivera and Vanneuville, in which a correlation
inequality was derived for certain topological events on the plane, as well as
by an old result of Piterbarg, in which a similar covariance formula was
established for finite-dimensional Gaussian vectors.