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Soft local times and decoupling of random interlacements. (English) Zbl 1329.60342
Summary: In this paper, we establish a decoupling feature of the random interlacement process \(\mathcal{I}^u \subset \mathbb Z^d\) at level \(u\), \(d \geq 3\). Roughly speaking, we show that observations of \(\mathcal{I}^u\) restricted to two disjoint subsets \(A_1\) and \(A_2\) of \(\mathbb Z^d\) are approximately independent, once we add a sprinkling to the process \(\mathcal{I}^u\) by slightly increasing the parameter \(u\). Our results differ from previous ones in that we allow the mutual distance between the sets \(A_1\) and \(A_2\) 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 \(u_{**}\), the probability of having long paths that avoid \(\mathcal{I}^u\) is exponentially small, with logarithmic corrections for \(d=3\).
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 on 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J55 Local time and additive functionals
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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