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Abstract
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We consider random interlacements on
,
, 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
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
is
not too large this asymptotic upper bound matches the asymptotic lower
bound derived by Sznitman [arXiv:1906.05809], and the exponential rate of
decay is governed by the variational problem in the continuum involving the
percolation function of the vacant set of random interlacements that he studied
[arXiv:1910.04737]. This is a further confirmation of the pertinence of this variational
problem.
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Keywords
large deviations, random interlacements, percolation
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Mathematical Subject Classification
Primary: 35A15, 60F10, 60K35, 82B43
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Milestones
Received: 9 September 2020
Revised: 15 February 2021
Accepted: 8 March 2021
Published: 15 October 2021
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