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Register a new userExistence of an unbounded vacant set for subcritical continuum percolation
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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 existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $mathbb{R}^d$, for any $dge2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.
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We consider the Bernoulli bond percolation process $mathbb{P}_{p,p}$ on the nearest-neighbor edges of $mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which this probability is $p$. Define [xi_{p,p}:=-lim_{ntoinfty}n^{-1}log mathbb{P}_{p,p}(0leftrightarrow nmathbf {e}_1)] and $xi_p:=xi_{p,p}$. We show that there exists $p_c=p_c(p,d)$ such that $xi_{p,p}=xi_p$ if $p<p_c$ and $xi_{p,p}<xi_p$ if $p>p_c$. Moreover, $p_c(p,2)=p_c(p,3)=p$, and $p_c(p,d)>p$ for $dgeq 4$. We also analyze the behavior of $xi_p-xi_{p,p}$ as $pdownarrow p_c$ in dimensions $d=2,3$. Finally, we prove that when $p>p_c$, the following purely exponential asymptotics holds: [mathbb {P}_{p,p}(0leftrightarrow nmathbf {e}_1)=psi_de^{-xi_{p,p}n}bigl(1+o(1)bigr)] for some constant $psi_d=psi_d(p,p)$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and dont rely on exact computations.
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 prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c e 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.
We consider the Brownian interlacements model in Euclidean space, introduced by A.S. Sznitman in cite{sznitman2013scaling}. We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in cite{benjamini2009visibility}. Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $pi_p(n) / pi_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at mesoscopic distance from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.
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