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For the Bargmann-Fock field on R d with d $ge$ 3, we prove that the critical level c (d) of the percolation model formed by the excursion sets {f $ge$ } is strictly positive. This implies that for every sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case = 0), {f = } contains an unbounded connected component that visits most of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behaviour of nodal hypersurfaces of these Gaussian fields in R d for d $ge$ 3 is very different from the behaviour of nodal lines of their two-dimensional analogues. Contents
We consider the Poisson Boolean percolation model in $mathbb{R}^2$, where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existen
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The purpose of this paper is extend recent results of Bonder-Groisman and Foondun-Nualart to the stochastic wave equation. In particular, a suitable integrability condition for non-existence of global solutions is derived.