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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.
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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.
Let $ mathbb{L}^{d} = ( mathbb{Z}^{d},mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ mathbb{Z}^{s} times { 0 }^{d-s} $, $ 2 leq s < d $, is open with probability $ q $ and every other edge is open with probability $ p $. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $ p eq p_{c}(d) $, where $ p_{c}(d) $ denotes the threshold for homogeneous percolation, and that the critical point $ (p,q_{c}(p)) $ can be approximated on the phase space by the critical points of slabs, for any $ p < p_{c}(d) $.
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.
One of the main objectives of equilibrium state statistical physics is to analyze which symmetries of an interacting particle system in equilibrium are broken or conserved. Here we present a general result on the conservation of translational symmetry for two-dimensional Gibbsian particle systems. The result applies to particles with internal degrees of freedom and fairly arbitrary interaction, including the interesting cases of discontinuous, singular, and hard core interaction. In particular we thus show the conservation of translational symmetry for the continuum Widom Rowlinson model and a class of continuum Potts type models.
A bootstrap percolation process on a graph $G$ is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infected and remains so forever. The parameter $rgeq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $beta$, where $2 < beta < 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)gg a_c(n)$, then there is a constant $eps>0$ such that, with high probability, the final set of infected vertices has size at least $eps n$. It turns out that when the maximum degree is $o(n^{1/(beta -1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Theta (n^{1/(beta -1)})$, then $a_c (n)=n^{beta -2 over beta -1}$.
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