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Register a new userInvariance under quasi-isometries of subcritical and supercritical behaviour in the Boolean model of percolation
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In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.
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We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.
We consider an anisotropic bond percolation model on $mathbb{Z}^2$, with $textbf{p}=(p_h,p_v)in [0,1]^2$, $p_v>p_h$, and declare each horizontal (respectively vertical) edge of $mathbb{Z}^2$ to be open with probability $p_h$(respectively $p_v$), and otherwise closed, independently of all other edges. Let $x=(x_1,x_2) in mathbb{Z}^2$ with $0<x_1<x_2$, and $x=(x_2,x_1)in mathbb{Z}^2$. It is natural to ask how the two point connectivity function $prob({0leftrightarrow x})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $prob({0leftrightarrow x})>prob({0leftrightarrow x})$. In this note we give an affirmative answer in the highly supercritical regime.
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.
We show that the quasi-stationary distribution of the subcritical contact process on $mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
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