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Register a new userCriticality and covered area fraction in confetti and Voronoi percolation
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Using the randomized algorithm method developed by Duminil-Copin, Raoufi, Tassion (2019b) we exhibit sharp phase transition for the confetti percolation model. This provides an alternate proof that the critical parameter for percolation in this model is $1/2$ when the underlying shapes for the distinct colours arise from the same distribution and extends the work of Hirsch (2015) and M{u}ller (2016). In addition we study the covered area fraction for this model, which is akin to the covered volume fraction in continuum percolation. Modulo a certain `transitivity condition this study allows us to calculate exact critical parameter for percolation when the underlying shapes for different colours may be of different sizes. Similar results are also obtained for the Poisson Voronoi percolation model when different coloured points have different growth speeds.
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We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $mathcal{P}_{lambda}$ in $mathbb{R}^2$ of intensity $lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(|x-y|)$, independent of everything else, where $g:[0,infty) to [0,1]$ and $| cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $mathcal{P}_{lambda}$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability $1 - expleft( - frac{eta W_xW_y}{|x-y|^{alpha}} right)$, $eta, alpha > 0$, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of $mathcal{P}_{lambda}$. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of $mathcal{P}_{lambda}$. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.
In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions where the number of strings distributed in the surface of interest is strongly determined by the size and the energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks and the shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems.
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.
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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