Journal of the European Mathematical Society

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Volume 17, Issue 10, 2015, pp. 2545–2593
DOI: 10.4171/JEMS/565

Published online: 2015-10-29

Soft local times and decoupling of random interlacements

Serguei Popov[1] and Augusto Teixeira[2]

(1) University of Campinas, Brazil
(2) IMPA, Rio de Janeiro, Brazil

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 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.

Keywords: Random interlacements, stochastic domination, soft local time, connectivity decay, smoothening of discrete sets

Popov Serguei, Teixeira Augusto: Soft local times and decoupling of random interlacements. J. Eur. Math. Soc. 17 (2015), 2545-2593. doi: 10.4171/JEMS/565