JournalsjemsVol. 17, No. 10pp. 2545–2593

Soft local times and decoupling of random interlacements

  • Serguei Popov

    University of Campinas, Brazil
  • Augusto Teixeira

    IMPA, Rio de Janeiro, Brazil
Soft local times and decoupling of random interlacements cover
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Abstract

In this paper we establish a decoupling feature of the random interlacement process IuZd\mathcal{I}^u \subset \mathbb Z^d at level uu, d3d \geq 3. Roughly speaking, we show that observations of Iu\mathcal{I}^u restricted to two disjoint subsets A1A_1 and A2A_2 of Zd\mathbb Z^d are approximately independent, once we add a sprinkling to the process Iu\mathcal{I}^u by slightly increasing the parameter uu. Our results differ from previous ones in that we allow the mutual distance between the sets A1A_1 and A2A_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 uu_{**}, the probability of having long paths that avoid Iu\mathcal{I}^u is exponentially small, with logarithmic corrections for d=3d=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 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. 2545–2593

DOI 10.4171/JEMS/565