Soft local times and decoupling of random interlacements

Conditional decoupling of random interlacements

“…We refer the reader to [13], Section 8, to see that A 1 possesses the desired properties. Note that, since s = o(r), the diameter of A 1 is of order r. We then define A 2 := A 2 (r, s) to be the set of points that are at least at distance 2s from A 1 :…”

“…In the present section we describe the technique introduced in [13], the so called Soft Local Times method. This method essentially allows us to simulate any number of random variables taking values in a state space Σ using a realization of a Poisson point process in Σ × R + .…”

“…This is what we are doing in this paper. To prove our results, the main method we use is a suitable modification (that allows dealing with conditional probabilities) of the soft local time method of [13]. We hope that this modification will be useful in other contexts, for instance, for dealing with the decoupling properties of the loop measures [3].…”

“…To complement the FKG inequality, we use sprinkling, i.e., we slightly change the intensity of random interlacements in order to decrease the error term; this approach was used in [17] and [18]. Then, in particular, in [13] it was proved that (1.5) It is important to observe, however, that the decoupling in the above form may not always be useful for one's needs. Intuitively, one is tempted to understand inequalities like (1.3) as "what happens in one set does not influence a lot what happens in the other set".…”

“…We prove a conditional decoupling inequality for the model of random interlacements in dimension d ≥ 3: the conditional law of random interlacements on a box (or a ball) A 1 given the (not very "bad") configuration on a "distant" set A 2 does not differ a lot from the unconditional law. The main method we use is a suitable modification of the soft local time method of [13], that allows dealing with conditional probabilities. …”