In this paper we establish a decoupling feature of the random interlacement process at level , . Roughly speaking, we show that observations of restricted to two disjoint subsets and of are approximately independent, once we add a sprinkling to the process by slightly increasing the parameter . Our results differ from previous ones in that we allow the mutual distance between the sets and to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold , the probability of having long paths that avoid is exponentially small, with logarithmic corrections for .
To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently from the rest of the paper.
Cite this article
Serguei Popov, Augusto Teixeira, Soft local times and decoupling of random interlacements. J. Eur. Math. Soc. 17 (2015), no. 10, pp. 2545–2593DOI 10.4171/JEMS/565