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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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