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Register a new userEuclidean and chemical distances in ellipses percolation
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The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $alpha$ controls the tail distribution of the major axis distribution and we focus on the regime $alpha in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between points $x$ and $y$ behaves roughly as $c loglog |x-y|$.
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We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $alpha > 0$ associated with the tail decay of the major axis distribution; we only consider distributions $rho$ satisfying $rho[r, infty) asymp r^{-alpha}$. We prove that this model presents a double phase transition in $alpha$. For $alpha in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $alpha in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $alpha in (2, infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $alpha = 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $mathbb{Z}^2$.
Scale-free percolation is a spatial random graph model with vertex set $mathbb{Z}^d$. Vertices $x$ and $y$ are connected with probability depending on i.i.d. vertex weights and the Euclidean distance. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances (in comparison to Euclidean distances). Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.
The non-random fluctuation is one of the central objects in first passage percolation. It was proved in [Shuta Nakajima. Divergence of non-random fluctuation in First Passage Percolation. {em Electron. Commun. Probab.} 24 (65), 1-13. 2019.] that for a particular asymptotic direction, it diverges in a lattice first passage percolation with an explicit lower bound. In this paper, we discuss the non-random fluctuation in Euclidean first passage percolations and show that it diverges in dimension $dgeq 2$ in this model also. Compared with the result in cite{N19}, the present result is proved for any direction and improves the lower bound.
We consider instances of long-range percolation on $mathbb Z^d$ and $mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $sin (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|toinfty$. For the model on $mathbb Z^d$ we show that, in probability as $|x|toinfty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(log r)^Delta$, where $Delta:=1/log_2(1/gamma)$ for $gamma:=s/(2d)$. For the model on $mathbb R^d$ we show that $D(0,xr)$ is, in probability as $rtoinfty$ for any nonzero $xinmathbb R^d$, asymptotic to $phi(r)(log r)^Delta$ for $phi$ a positive, continuous (deterministic) function obeying $phi(r^gamma)=phi(r)$ for all $r>1$. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.
We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for the multi-type Richardson model. In two dimensions this establishes a dual relation between the existence of infinite geodesics and coexistence among competing types. The argument amounts to making precise the heuristic that infinite geodesics can be thought of as `highways to infinity. We explain the limitations of the current techniques by presenting a partial result in dimensions higher than two.
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