Excess deviations for points disconnected by random interlacements

Abstract

We consider random interlacements on ${\mathbb{Z}}^d$, $d\ge 3$, when their vacant set is in a strongly percolative regime. Given a large box centered at the origin, we establish an asymptotic upper bound on the exponential rate of decay of the probability that the box contains an excessive fraction $\nu$ of points that are disconnected by random interlacements from the boundary of a concentric box of double size. As an application we show that when $\nu$ is not too large, this asymptotic upper bound matches the asymptotic lower bound derived in a previous work of the author, and the exponential rate of decay is governed by a certain variational problem in the continuum which involves the percolation function of the vacant set of random interlacements.

46 pages, 3 figures, to appear in Probability and Mathematical Physics

References in corpus (2)

• On the uniqueness of the infinite cluster of the vacant set of random interlacements
• Equality of critical parameters for percolation of Gaussian free field level-sets

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• Analyticity of Gaussian free field percolation observables
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• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$